The gradient discretisation method

Transcription

1 The gradient discretisation method Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaele Herbin To cite this version: Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard, Raphaele Herbin. The gradient discretisation method : A framework for the discretisation and numerical analysis of linear and nonlinear elliptic and parabolic problems <hal v3> HAL Id: hal Submitted on 1 Nov 216 (v3), last revised 9 Jul 218 (v7) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin The gradient discretisation method A framework for the discretisation and numerical analysis of linear and nonlinear elliptic and parabolic problems November 1, 216

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4 Preface This monograph is dedicated to the presentation of the Gradient Discretisation Method (GDM) and of some of its applications. It is intended for masters students, researchers and experts in the field of the numerical analysis of partial differential equations. The GDM is a framework which contains classical and recent discretisation schemes for diffusion problems of different kinds: linear or non linear, steadystate or time-dependent. The schemes may be conforming or non conforming and may rely on very general polygonal or polyhedral meshes. In this monograph, the core properties that are required to prove the convergence of a GDM are stressed, and the analysis of the method is performed on a series of elliptic and parabolic problems, linear or non-linear, for which the GDM is particularly adapted. As a result, for these models, any scheme entering the GDM framework is known to converge. A key feature of this monograph is the presentation of techniques and results which enable a complete convergence analysis of the GDM on fully non-linear, and sometimes degenerate, models. The scope of some of these techniques and results goes beyond the GDM, and makes them potentially applicable to numerical schemes not (yet) known to fit into this framework. Appropriate tools are then developed so as to easily check whether a given scheme satisfies the expected properties of a GDM. Thanks to these tools a number of methods can be shown to enter the GDM framework: some of these methods are classical, such as the conforming Finite Elements, the Raviart Thomas Mixed Finite Elements, or the P 1 non-conforming Finite Elements. Others are more recent, such as the Hybrid Mixed Mimetic or Nodal Mimetic methods, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes. Marseille, Melbourne, Paris the authors, 216

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6 Contents Part I Elliptic problems 1 Motivation and basic ideas Some well-known approximations of linear elliptic problems Galerkin approximations Non-conforming P 1 finite elements Two-point flux finite volume on Cartesian meshes Towards Gradient Schemes Generalisation to non-linear problems The gradient discretisation method Dirichlet boundary conditions Homogeneous Dirichlet conditions Non-homogeneous Dirichlet boundary conditions Neumann boundary conditions Homogeneous Neumann conditions Non-homogeneous Neumann conditions Complements on trace operators Non-homogeneous Fourier boundary conditions Mixed boundary conditions Elliptic problems The linear case Homogeneous Dirichlet boundary conditions Non-homogeneous Dirichlet boundary conditions Neumann boundary conditions Mixed boundary conditions Unknown-dependent diffusion problems Homogeneous Dirichlet boundary conditions Non-homogeneous Dirichlet boundary conditions Homogeneous Neumann boundary conditions

7 VIII Contents Non-homogeneous Neumann boundary conditions Non-homogeneous Fourier boundary conditions p-laplacian type problems: p (1, + ) An error estimate for the p-laplace problem Convergence of gradient schemes for fully nonlinear Leray Lions problems Part II Parabolic problems 4 Time-dependent problems: GDM and DFA Space time gradient discretisation Averaged-in-time compactness Abstract setting Application to space time gradient discretisations Uniform-in-time compactness Definitions and abstract results Application to space time gradient discretisations Parabolic problems The gradient discretisation method for a quasilinear parabolic problem The continuous problem The gradient scheme Error estimate in the linear case Convergence analysis in the non-linear case Non-conservative problems The continuous problem Fully implicit scheme Semi-implicit scheme Non-linear time-dependent Leray Lions problems Model Gradient scheme and main results A priori estimates Proof of the convergence results Degenerate parabolic problems The continuous problem Hypotheses and notion of solution A maximal monotone operator viewpoint Gradient scheme Estimates on the approximate solution A first convergence theorem Uniform-in-time, strong L 2 convergence results Auxiliary results

8 Contents IX 6.7 Proof of the uniqueness of the solution to the model Numerical example Part III Review of gradient discretisation methods 7 Meshes and discrete tools Polytopal meshes Definition and notations Operators, norm and regularity factors associated with a polytopal mesh Polytopal toolboxes Dirichlet boundary conditions Non-homogeneous Dirichlet boundary conditions Neumann and Fourier boundary conditions Mixed boundary conditions Local linearly exact GDs P -exact and P 1 -exact reconstructions Definition and consistency of local linearly exact GDs for Dirichlet boundary conditions From local to global basis functions, and matrix assembly Barycentric elimination of degrees of freedom Mass lumping Non-homogeneous Dirichlet, Neumann and Fourier boundary conditions Conforming methods and derived methods Conforming Galerkin methods Homogeneous Dirichlet boundary conditions Non-homogeneous Neumann boundary conditions P k finite elements for homogeneous Dirichlet boundary conditions Definition of P k gradient discretisations Properties of P k gradient discretisations P k FE for non-homogeneous Dirichlet, Neumann and Fourier BCs Non-homogeneous Dirichlet conditions Neumann boundary conditions Fourier conditions Mass-lumped P 1 finite elements Vertex approximate gradient (VAG) methods

9 X Contents 9 Non-conforming finite element methods and derived methods Non-conforming P 1 FE for homogeneous Dirichlet BCs Definition of the non-conforming P 1 gradient discretisation Preliminary lemmas Properties of the non-conforming P 1 finite element method Non-conforming P 1 methods for Neumann and Fourier BCs Neumann boundary conditions Fourier boundary conditions Non-conforming P 1 FE for non-homogeneous Dirichlet BCs Mass-lumped non-conforming P 1 reconstruction Mixed finite element RT k schemes The RT k mixed finite element scheme for linear elliptic problems Gradient discretisation from primal mixed finite element Gradient discretisation from dual mixed finite element formulation The multi-point flux approximation MPFA-O scheme MPFA methods for Dirichlet boundary conditions Definition of the MPFA gradient discretisation Preliminary lemmas Properties of the MPFA-O gradient discretisation MPFA-O methods for Neumann and Fourier boundary conditions Neumann boundary conditions Fourier boundary conditions Hybrid mimetic mixed schemes HMM methods for Dirichlet boundary conditions Definition of HMM gradient discretisations Preliminary lemmas Properties of HMM gradient discretisations HMM methods for Neumann and Fourier boundary conditions Neumann boundary conditions Fourier boundary conditions HMM fluxes, and link with the two-point finite volume method A cell-centred variant of HMM schemes on -admissible meshes The SUSHI scheme for homogeneous Dirichlet conditions Harmonic interpolation coefficients

10 Contents XI 13 Nodal mimetic finite difference methods Definition and properties of nmfd gradient discretisations Preliminary lemmas Properties of nmfd gradient discretisations Link with discrete duality finite volume methods Part IV Appendix A Complements on LLE GDs A.1 W 2,p estimates for S D A.2 LLE GDs with generalised degrees of freedom A.3 Non-linearly exact barycentric combinations B Discrete functional analysis B.1 Preliminary results B.1.1 Geometrical properties of cells B.1.2 Approximation properties B.2 Discrete functional analysis for Dirichlet boundary conditions. 383 B.2.1 Discrete Sobolev embeddings B.2.2 Compactness in L p () B.3 Discrete functional analysis for Neumann and Fourier BCs B.3.1 Estimates involving the reconstructed trace B.3.2 Discrete Sobolev embeddings B.3.3 Compactness in L p () B.4 Discrete functional analysis for mixed boundary condition B.4.1 Discrete Sobolev embeddings B.4.2 Compactness in L p () C Technical results C.1 Standard notations, inequalities and relations C.1.1 Notations C.1.2 Hölder inequalities C.1.3 Young inequality C.1.4 Jensen inequality C.1.5 Power of sums C.1.6 Discrete integration-by-parts (summation-by-parts) C.2 Topological degree C.3 Weak and strong convergences in integrals C.4 Minty trick and convexity inequality References

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12 Introduction The purpose of this book is the study of the Gradient Discretisation Method (GDM), which includes a large family of conforming or non conforming numerical methods for diffusion problems. A Gradient Discretisation Method is based on the choice of a set of discrete spaces and operators, referred to as a Gradient Discretisation (GD). Using the discrete elements of a particular GD in lieu of continuous space and operators in the weak formulation of a diffusion problem then yields a numerical scheme called a Gradient Scheme (GS) for this problem. Considering here only the case of homogeneous Dirichlet boundary conditions, the stationary linear and non-linear diffusion problems that we shall consider can be written under the form: where div a(x, u, u) = f in, u = on, is an open bounded connected subset of R d, d N, with a Lipschtizregular boundary denoted by = \, a is a function from R d R R d to R d. The function a may be a general anisotropic heterogeneous linear operator, that is a(x, u, ξ) = Λ(x)ξ, which yields a linear diffusion problem. Another possible choice for a is a Leray-Lions operator such as the p-laplacian a(x, u, ξ) = ξ p 2 ξ with p > 1, which yields a non linear diffusion problem. We also consider the related evolution problem: t u div a(x, u, u) = f in (, T ), u(x, ) = u ini (x) in, u = on (, T ). Degenerate transient problems are also treated, such as the following:

13 2 Contents t β(u) ζ(u) = f in (, T ), β(u(x, )) = β(u ini (x)) in, ζ(u) = on (, T ), where the functions ζ and β will mainly be assumed to be Lipschitz-continuous and non-decreasing. This model includes both the Stefan problem arising from a melting material model, and the Richards problem which models a two phase flow in a porous media problem under the assumption that the pressure of one of the phases is given. The above problems arise in various frameworks, such as underground engineering (oil recovery, hydrology, nuclear waste disposals, etc.), or in image processing. In the case of underground engineering, numerical simulations have to be performed on meshes adapted to the geological layers, which include complex geometrical features such as faults, vanishing layers, inclined wells, highly heterogeneous permeability fields, local non conforming refinement. A large number of discretisation methods have recently been developed for the numerical approximation of these equations; we show in this book that several of these recent methods, and many of the more classical ones, are GDMs, in particular: 1. the conforming or non-conforming Finite Element methods, including mass lumped versions, 2. the Raviart-Thomas Mixed Finite Element methods, 3. the Multi-Point Flux Approximation (MPFA) schemes and the Discrete Duality Finite Volume (DDFV) schemes on particular grids, 4. the Hybrid Mimetic Mixed (HMM) family which includes the hybrid Mimetic Finite Difference schemes, the SUSHI scheme and the Mixed Finite Volume scheme, 5. the nodal Mimetic Finite Difference scheme. This book is written assuming that the reader is familiar with Sobolev spaces and weak formulations of elliptic and parabolic partial differential equations. We refer to [13] for an introduction on this topic. The reader should also have some notions of numerical analysis, in particular of the discretisation of elliptic and parabolic partial differential equations (PDEs) : for example the knowledge of one of the aforementioned methods (such as the conforming P 1 Finite Elements on triangles). This book is organised as follows. In Part I, we first motivate in Chapter 1 the basic concepts used to define a GDM. This method is then formally introduced in Chapter 2. Chapter 3 shows how the GDM is applied to elliptic problems. An error estimate is first obtained in the linear case. The convergence of the GDM is then proved for a generalised Leray Lions model, which involves a non-local dependency of the operator. Part II is devoted to the study of the GDM for linear and non-linear parabolic problems. In Chapter 4, we present the definitions and main compactness results which are used to analyse the GDM for nonlinear parabolic problems. In

14 Contents 3 Chapter 5, we start with the classical parabolic heat equation, and then study the convergence of gradient schemes for transient Leray-Lions problem. Chapter 6 covers the study of GSs for degenerate parabolic problems, including the Stefan problem and the Richards problem. We also note that, even though numerous kinds of equations are covered here, they do not form an exhaustive list of the models for which gradient schemes have been developed and analysed. In particular, the following models are not covered in this monograph: linear and non-linear elasticity equations [41], the poro-elasticity equations [63], the Stokes and Navier Stokes equations [36, 57], and the obstacle and Signorini problems [3, 4]. Part III lists some important examples of GDMs. We first show that the standard Finite Elements Methods, the Vertex Approximate Gradient scheme, the non-conforming P 1 Finite Elements Method and the Mixed Finite Elements Methods are GDMs. We then analyse, in the framework of GDM, the HMM family, some particular Finite Volume methods (MPFA, DDFV...), and the nodal Mimetic Finite Difference method. In the appendix, we provide tools to establish that particular gradient discretisations satisfy the required properties for the convergence analysis of Parts I and II to hold. In particular, the following notions are introduced in Chapter A: - local linearly exact gradient reconstruction, which rigorously describes the basic idea of numerous schemes for diffusion equations, - barycentric elimination, which enables a reduction of the number of degrees of freedom of a method; - mass-lumping, particularly useful to deal with time-dependent or some non-linear models. The second chapter of the appendix, Chapter B, describes further tools useful to analyse GDs. These discrete functional analysis tools are gathered into the notion of polytopal toolbox, which is relevant to a wide range of gradient discretisations. Let us conclude this introduction by mentioning that some of the techniques developed in Chapters 3, 5 and 6 (convergence analysis for fully non-linear and degenerate models), Chapter 4 (compactness theorems for time-dependent problems), and Chapter B in the appendix (discrete Sobolev and Rellich embeddings) are potentially applicable outside the GDM, to schemes that are not necessarily written as gradient schemes. Remark.1 (Shaded remarks) Shaded remarks such as this one contains notions, comments or results that can be somewhat technical, and can be skipped in a first reading.

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16 Part I Elliptic problems

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18 1 Motivation and basic ideas 1.1 Some well-known approximations of linear elliptic problems Let us consider the following simple elliptic problem: { u = f in, u = on, (1.1) where is a polygonal subset of R d and f L 2 (). The weak formulation of (1.1) is: Find u H 1 () such that, for all v H 1 (), (1.2) u(x) v(x)dx = f(x)v(x)dx Galerkin approximations A classical family of numerical methods to approximate this problem is given by conforming Galerkin methods: the main idea is to seek the approximate solution in a finite dimensional subspace V h of H 1 (). This is for example the case for the well-known P 1 finite element method, in which a partition of into simplices (e.g. triangles in dimension d = 2) is chosen and the space V h is made of the piecewise linear functions on this partition, which are continuous over and have a zero value on. In such a case, the index h denotes the mesh size, see e.g. [22] for more on Finite Element approximations. Once such a finite dimensional subspace V h has been chosen, the Galerkin approximation of (1.2) is Find u h V h such that, for all v h V h, (1.3) u h (x) v h (x)dx = f(x)v h (x)dx.

19 8 1 Motivation and basic ideas It is then easy to establish an error bound between the weak solution u to (1.1) and the approximate solution u h. Using a generic v = v h V h H 1 () as a test function in (1.2) and subtracting (1.3) we see that (u u h )(x) v h (x)dx =. (1.4) Taking v h = w h u h where w h is any function in V h and writing v h = w h u + u u h gives (u u h )(x) (u u h )(x)dx = (u u h )(x) (u w h )(x)dx. Using Cauchy-Schwarz inequality in the right-hand side (see Section C.1) and recalling that ϕ 2 H 1 () = ϕ(x) 2 dx, we infer that u u h 2 H 1 () u u h H 1 () u w h H 1 (). Finally, since this estimate is valid for any w h V h, u u h H 1 () min w h V h w h u H 1 (). (1.5) This result may be generalised to the case of a bilinear form a and is known as Céa s lemma [21]. If a family of subspaces (V h ) h> is such that V h becomes ultimately dense in H 1 () as h, i.e. for all ϕ H 1 (), min wh V h w h ϕ H 1 () as h, then Estimate (1.5) shows that u h u in H 1 () as h. The beauty of this analysis is its simplicity. It is however limited to methods for which the approximation space V h is included in the space V in which the continuous solution lives. These methods are refered to as conforming. Numerous numerical schemes for elliptic equations are not conforming in the sense that the provided approximate solutions are not in H 1 (). This happens for instance in the case of the non-conforming P 1 finite element, which yields a piecewise affine approximation and in the case of the cell-centred finite volume scheme, which yields a piecewise constant approximation Non-conforming P 1 finite elements We again consider a simplicial mesh M of a domain. Let F be the finite set of the faces of the mesh (edges in 2D), F ext be the set of all σ F such that σ, and F int = F \ F ext the set of interior faces. For any σ F, x σ is the barycenter (center of mass) of σ. The approximation space V h of the nonconforming P 1 finite element method is the set of piecewise affine functions on the simplices of the mesh; such a function is depicted in Figure 1.1. The space V h is spanned by the basis (ϕ σ ) σ Fint, where ϕ σ is the piecewise affine

20 1.1 Some well-known approximations of linear elliptic problems 9 u h x σ K σ Fig Non-conforming P 1 finite element, two-dimensional case function such that ϕ σ (x σ ) = 1 and ϕ σ (x σ ) = for all σ F \{σ}. The space V h is clearly not a subspace of H 1 (); however, the restriction of a function of V h is piecewise affine so that its gradient is well defined and constant. For K M, let us denote by K ϕ σ the constant value of the gradient of the function ϕ σ, σ F, on K (note that K ϕ σ = if σ is not an interface of K. It is remarkable that (1.3) still makes sense if the gradient operator in this formula is replaced by the broken gradient operator M defined by For any u h = u σ ϕ σ, σ F int K M, x K, M u h (x) = σ F int u σ K ϕ σ (1.6) (in other words, the gradients are computed without taking into account the jump along the edges). Then the following norm is defined on H 1 () + V h : u h H 1 () + V h, u h 2 h = u h (x) 2 dx. K M If u h V h, then we have u h h = M u h L 2 () ; if u h H 1 (), then we have u h h = u h L2 (). In order to approximate Problem (1.1), we define the bilinear form a h : (H 1 () + V h ) 2 R by (u h, v h ) (H 1 () + V h ) 2, a h (u h, v h ) = u h (x) v h (x)dx. K K M We see that a h is elliptic with an ellipticity constant α = 1, since K v h V h, a h (v h, v h ) = v h 2 h. The approximate solution of Problem (1.1) is defined by

21 1 1 Motivation and basic ideas Find u h V h such that, v h V h, a h (u h, v h ) = f(x)v h (x)dx. (1.7) There exists one and only one solution to (1.7), and there holds the following error estimate [21, Theorem 4.2.2], based on the second Strang Lemma [69]: there exists C >, depending only on the regularity of M but not on h, such that u u h h C inf u v h v h V h + h sup w h V h \{} a h(u, w h ) f(x)w h (x)dx w h. (1.8) h This estimate can be written in terms solely involving u: u u h h C(S M (u) + W M ( u)), (1.9) where S M (ϕ) is defined, for any ϕ H 1 (), by S M (ϕ) = inf ϕ v h h, (1.1) v h V h and W M (ϕ) is defined, for any sufficiently regular function ϕ : R d, by (ϕ(x) M w h (x) + divϕ(x)w h (x))dx W M (ϕ) =. (1.11) w h h sup w h V h \{} Under regularity assumptions on the mesh, the quantities, S M (ϕ) and W M (ϕ) tend to zero as the size of the mesh tends to zero, see e.g. [22, 45] Two-point flux finite volume on Cartesian meshes A second example of a non-conforming scheme is given by the Two-Point Flux Approximation (TPFA) finite volume scheme [49]. The TPFA scheme on Cartesian grids, which we shall denote by TPFA-CG is widely used in petroleum engineering: constant values are considered in control volumes over which a discrete mass balance of the various components is established. Let us consider a rectangular mesh of a rectangle (d = 2). In addition to the notations K, σ and x σ introduced above, we introduce the following (see Figure 1.2): We denote by x K the centre of mass of K M, by V the set of vertices of the mesh, by V K the set of vertices of K, and by F K the set of the edges of K.

22 1.1 Some well-known approximations of linear elliptic problems 11 σ u σ s u K V K,s σ u σ L x K x σ x L K n K,σ Fig Notation for a rectangular mesh. For each K M and each s V K, V K,s is the rectangle defined by x σ, s, x σ and x K, where σ and σ are the edges of K touching s. u K (resp. u σ ) represents an approximate value of the unknown u at x K (resp. x σ ). The idea of finite volume schemes consists in finding approximate values F K,σ of the exact fluxes σ u n K,σds(x) (n K,σ is the normal to σ outward K, ds is the measure on the edges), and in writing the following discrete flux balance in each cell K M, F K,σ = f(x)dx, (1.12) σ F K K and flux conservativity across each interior edge: σ F int common face of K and L, F K,σ + F L,σ =. (1.13) Relation (1.12) simply mimicks the Stokes formula applied to the continuous problem (1.1): u n K,σ ds(x) = f(x)dx. σ F K σ The TPFA-CG finite volume scheme consists in substituting, in the previous equations, u σ u K F K,σ = σ dist(x σ, x K ). (1.14) The boundary condition is imposed by setting K u σ = if σ, (1.15)

23 12 1 Motivation and basic ideas There is no clear way to see the TPFA-CG scheme method as a nonconforming finite element method. However, it can be recast into a variational formulation. We first consider a family ((v K ) K M, (v σ ) σ F ) such that v σ = if σ. Multiplying (1.12) by v K and summing on K M, we get v K F K,σ = v K f(x)dx. (1.16) K M σ F K K M K We then notice that K M σ F K v K F K,σ = K M σ F K (v K v σ )F K,σ. (1.17) Indeed, if σ, then (v K v σ )F K,σ = v K F K,σ. If σ is the common face between two control volumes K and L, then v σ is multiplied in the above sum by F K,σ + F L,σ, which vanishes thanks to (1.13). Thus, using (1.17) into (1.16) and invoking (1.14), we get K M σ F K σ dist(x σ, x K ) (v σ v K )(u σ u K ) = K M v K K f(x)dx. (1.18) Conversely, it is possible, assuming Expression (1.14) for the fluxes, to deduce (1.12), (1.13) and (1.15) from (1.18) with convenient choices for the family ((v K ) K M, (v σ ) σ F ) (namely, selecting only one value equal to 1 and all the other ones equal to ). Moreover, Relation (1.18) can be expressed in terms of reconstructed functions and gradients, using the discrete values defined on K and σ. We define X D, as the space of all real families u D = ((u K ) K M, (u σ ) σ F ) satisfying the boundary conditions(1.15). For u D X D,, we let Π D u D be the piecewise constant function equal to u K on the cell K. If K M and s V K is such that σ and σ are the faces of K sharing the vertex s, we define the reconstructed gradient K,s u D = (σ) K,s u D n K,σ + (σ ) K,s u D n K,σ with (σ) K,s u D = u σ u K dist(x σ, x K ) and (σ ) K,s u D = u σ u K dist(x σ, x K ). We then denote by D u D the piecewise constant function equal to K,s u D on V K,s, for any cell K and any vertex s V K. The following properties arise, for (u D, v D ) X 2 D, : K M v K K f(x)dx = f(x)π D v D (x)dx,

24 and K M σ F K 1.1 Some well-known approximations of linear elliptic problems 13 σ dist(x σ, x K ) (v σ v K )(u σ u K ) = D u D (x) D v D (x)dx. As a result, we can rewrite (1.18) in the form of a discrete variational problem: Find u D X D, such that, for all v D X D,, (1.19) D u D (x) D v D (x)dx = f(x)π D v D (x)dx. The study of the TPFA-CG scheme has been done in [49] using finite volume techniques, and the following results are proven: if the size of the mesh tends to, then Π D u D converges to u in L 2 (), and an error estimate holds, which depends on the regularity of u. Remark 1.1 (Unstructured meshes). The analysis of the TPFA scheme of [49] also holds in the case of unstructured meshes, provided an orthogonality condition holds (see [49, Definition 9.1]). However, in the unstructured case, it does not seem to be possible to write the scheme under the form (1.19), except in the so-called super-admissible case, i.e. when the centre of mass is also the intersection of the orthogonal bisectors; therefore, the general TPFA scheme is not included in the framework of the gradient schemes studied in this book. The question arises to know whether the TPFA-CG scheme could be studied using the non-conforming techniques of Section There are a series of objections to this approach: 1. Comparing the right-hand sides of (1.7) and (1.19) we see that the natural space V h would be V h = {Π D v D, v D X D, }. However, this space forgets about the edge degrees of freedom (v σ ) σ F of v D X D,, and there is thus no way to compute D v D solely from Π D v D. 2. Partially as a consequence of the previous item, there does not seem to exist any bilinear form a h, defined on V h + H 1 (), which gives back Du D (x) D v D (x)dx for elements of V h, and u(x) v(x)dx for elements of H 1 (). 3. The same problem arises for the definition of the norm h. Although we cannot directly use on the TPFA-CG scheme the technique from non-conforming finite elements schemes, there is however a way of merging these two kinds of schemes into a common framework, which also covers conforming finite element methods. The next section presents an introduction to this framework.

25 14 1 Motivation and basic ideas 1.2 Towards Gradient Schemes What does it take to design a unified convergence analysis framework covering the preceding three examples, as well as other conforming and non-conforming methods? A numerical method obviously starts from selecting a finite number of degrees of freedom describing the finite dimensional space in which the approximate solution is sought. We already called X D, this finite dimensional space ( D for discretisation, and the to indicate that, in some way, this space accounts for the homogeneous boundary condition in (1.1)). The two linear operators Π D and D, which respectively reconstruct, from the degrees of freedom, a function on and its gradient, are such that Π D : X D, L 2 () and D : X D, L 2 () d. All the schemes presented in the previous section can be written as Find u D X D, such that, for all v D X D,, D u D (x) D v D (x)dx = f(x)π D v D (x)dx (1.2) for suitable choices of (X D,, Π D, D ). For conforming P 1 finite elements, each v D X D, is a vector of values at the vertices of the mesh, Π D v D C() is the piecewise linear function on the mesh which takes these values at the vertices, and D v D = (Π D v D ). For non-conforming P 1 elements, each v D X D, is a vector of values at the barycenters of the edges, Π D v D is the piecewise linear function on the mesh which takes these values at these barycenters, and D v D = M (Π D v D ) is the broken gradient defined in (1.6). The space and operators for the TPFA-CG scheme have already been given under the form (X D,, Π D, D ) in the previous section. The question now is to understand which properties the triplet (X D,, Π D, D ) must satisfy to enable some error estimates between the solution u to (1.2) and the solution u D to (1.2) (assuming for the time being that it exists). The main issue is that, contrary to Problem (1.2) and its conforming discretisation (1.3), Problem (1.2) and its general discretisation (1.2) do not appear to have any common test functions. Hence, no equation equivalent to (1.4) seems attainable. There is however a way to write an approximate version of this relation in the same spirit as in the analysis of the nonconforming finite element method; however, we no longer assume that a bilinear form a h or a norm h can be defined with argument Π D u; as mentioned above, this is mandatory if we want to include the TPFA-CG scheme in this framework. By noticing that (1.2) implies that u = f in the sense of distributions, we get from (1.2) that, for any v D X D,, D u D (x) D v D (x)dx = u(x)π D v D (x)dx. (1.21)

26 1.2 Towards Gradient Schemes 15 If Π D v D were a classical regular function, Stokes formula would allow us to replace the integrand in the right-hand side with u(x) (Π D v D )(x). Except in some particular cases, the discrete operators Π D, D of a numerical scheme do not satisfy an exact discrete Stokes formula, only an approximate one. We measure the resulting defect of conformity of the method, in the spirit of (1.11), by a function W D (ϕ) such that, for any sufficiently regular function ϕ : R d, and for all v D X D,, W D (ϕ) = sup v D X D, (ϕ(x) D v D (x) + divϕ(x)π D v D (x)dx)dx D v D L 2 () d. (1.22) Here, we assume that D v D L2 () if v d D, which is somewhat natural given the homogeneous boundary conditions we come back to this further down. The quantity W D (ϕ) is expected to be small if the discretisation is fine enough (e.g. the underlying mesh size is small). Then, considering ϕ = u in (1.22) and using (1.21) to compute u(x)π Dv D (x)dx, we obtain an approximate version of (1.4): ( u(x) D u D (x)) D v D (x)dx D v D L2 () W D( u). d We now take a generic w D X D,, apply this estimate to v D = w D u D, and write u D u D = u D w D + D w D D u D to find ( D w D (x) D u D (x)) ( D w D (x) D u D (x))dx ( D w D (x) u(x)) ( D w D (x) D u D (x))dx + D (w D u D ) L2 () d W D( u). Using the Cauchy-Schwarz inequality on the first term in the right-hand side, we infer D u D D w D L2 () d u Dw D L2 () d + W D( u). (1.23) We now define the best interpolation error (in the spirit of (1.1)) by ( ) S D (u) := min Π D w D u w D X L2 () + Dw D u L2 (). d D, We pick w D which realises this minimum. Since D u D u L 2 () Du d D D w D L 2 () + Dw d D u L 2 () d Equation (1.23) gives D u D D w D L2 () d + S D(u),

27 16 1 Motivation and basic ideas D u D u L 2 () d 2S D(u) + W D ( u). (1.24) Let us now study how Π D u D discrete Poincaré inequality: approximates u. To this aim, we assume a There exists C D > such that, v D X D,, This enables us to write Π D v D L2 () C D D v D L2 () d. Π D u D u L 2 () Π Du D Π D w D L 2 () + Π Dw D u L 2 () C D D u D D w D L2 () d + S D(u). Estimate (1.23) then shows that Π D u D u L2 () (C D + 1)S D (u) + C D W D ( u). (1.25) Equations (1.24) and (1.25) are error estimates between u and Π D u D and between u and D u D. In particular, if we take a sequence (X Dm,, Π Dm, Dm ) m N such that (P1) (C Dm ) m N is bounded, (P2) S Dm (u) as m, (P3) W Dm ( u) as m, then (1.24) and (1.25) show that, as m, Π Dm u Dm u in L 2 () and that Dm u Dm u in L 2 () d. The previous reasoning highlights the core properties that (X D,, Π D, D ) must satisfy to provide a proper approximation of (1.1) under the form (1.2). Property (P1) is related to some coercivity property of this triplet, since this uniform Poincaré inequality is also what ensures an estimate of the form D u D L2 () C f L 2 () if u D is a solution to (1.2). Property (P2) states that Π D and D are consistent reconstructions of functions and their gradient; it allows us to approximate u and its gradient by using elements in X D,. As already discussed, W easures the error in the discrete Stokes formula and (P3) therefore relates to the limit-conformity of (Π D, D ), stating that these two operators must ultimately behave as in the conforming case and should, in the limit, satisfy the exact Stokes formula. 1.3 Generalisation to non-linear problems The framework of the convergence analysis must be able to handle non-linear equations. Assume we now wish to approximate the following problem:

28 1.3 Generalisation to non-linear problems 17 { β(u) u = f in, u = on, (1.26) with the same notations as in Section 1.1, and where the function β : R R is continuous. The weak formulation of (1.26) is: Find u H 1 () such that, for all v H 1 (), (1.27) (β(u(x))v(x) + u(x) v(x))dx = f(x)v(x)dx. It can then be shown that there exists at least one solution to (1.27). Let us assume that we use a conforming Galerkin method, say the P 1 finite element method. Denoting by V h the space of continuous piecewise linear functions on a triangular mesh of, the P 1 finite element approximation of (1.27) could be Find u h V h such that, for all v h V h, (1.28) (β(u h (x))v h (x) + u h (x) v h (x))dx = f(x)v h (x)dx. Although this approximate problem has at least one solution, its analysis presents three major difficulties. The first one is the difficulty to compute, if u h = s V int u s ϕ s, the quantity ( ) β u s ϕ s (x) ϕ s (x)dx (1.29) s V int for a given interior vertex s of the mesh; due to the non-linearity β, the integrand may not be a piecewise polynomial and thus exact quadrature rules cannot be used. The second one is to define an algorithm to approximatie the solution of the non-linear system of equations provided by (1.28). The third one is to prove that the numerical method converges to the solution of the initial problem. A classical answer to the first issue is to use the so-called mass-lumping method. This method consists in replacing, in (1.28) with v h = ϕ s, the term (1.29) with ω s β(u s ) where ω s is some weight to be defined. The gradient scheme framework provides a natural way of analysing the stability and convergence of this mass-lumped scheme, with weights defined as the measure of some dual cells denoted by K s (see figure 1.3). We simply let X D, = R Vint as before and we define, for u X D,, Π D u = u s 1 Ks (piecewise constant reconstruction), s V int D u = (1.3) u s ϕ s, s V int where 1 Ks is the characteristic function of K s. Then the scheme (1.28) is replaced with

29 18 1 Motivation and basic ideas K s s Fig Definition of K s Find u X D, such that, for all v X D,, (β(π D u(x))π D v(x) + D u(x) D v(x))dx = f(x)π D v(x)dx. (1.31) We first notice that, since Π D u and Π D v are piecewise constant, this scheme leads to a very simple computation of both the first term in the left-hand side and the term in the right-hand side. Another major interest of dealing with these piecewise constant reconstructions Π D is that they satisfy β(π D u) = Π D (β(u)) = s V int β(u s )1 Ks. This is of crucial importance for the obtention of the estimates (see section 6.3). It is also important from a numerical point of view since it facilitates the implementation of the scheme. If (X Dm,, Π Dm, Dm ) m N is a sequence of mass-lumped P 1 finite elements it is possible to show that properties (P1), (P2) and (P3) as defined at the end of Section 1.2 hold, provided that they hold for the underlying P 1 finite elements. Then, only using those properties, it is possible to show that: 1. The scheme (1.31) has at least one solution, which we denote by u m X Dm,, 2. Up to a subsequence, as m, Π Dm u m converges weakly in L 2 () to some function u H 1 (), Dm u m converges weakly in L 2 () d to u, and β(π Dm u m ) converges weakly in L 2 () to some function β. It is however not possible in general to deduce from Properties (P1) (P3) that β = β(u). An additional compactness property of the sequence (X Dm,, Π Dm,

30 1.3 Generalisation to non-linear problems 19 Dm ) m N is necessary to prove that β = β(u), and thus complete the convergence proof of the scheme: (P4) for any bounded sequence ( Dm u m ) m N, (Π Dm u m ) m N is relatively compact in L 2 (). This property can be established for gradient schemes defined by the masslumped P 1 finite element method. The discrete elements (X D,, Π D, D ) and Properties (P1), (P2), (P3), (P4) are at the core of the definition and properties of the gradient discretisation method (GDM), which provide the foundations to build gradient schemes, in addition with the piecewise constant reconstruction property in the case of some nonlinear problems (see (1.3)).

31

32 2 The gradient discretisation method A gradient discretisation method (GDM) is a numerical technique used to find approximate solutions to boundary value problems for elliptic and parabolic partial differential equations. As mentioned in its name, the GDM relies on a gradient discretisation (GD), denoted by D, which contains at least the three following discrete entities: a discrete space of unknowns X D is a, e.g. the values at the nodes of the mesh, as in the conforming P1 finite element method, at a particular point of the cell, as in the TPFA-CG scheme, or at a particular point of the faces, as in the non-conforming P1 finite element method), a function reconstruction operator Π D which transforms an element of X D into a function defined a.e. on the physical domain. D is a an approximate gradient reconstruction, which builds a discrete gradient (vector-valued function) defined a.e. on from the discrete unknowns. In the present chapter, we define the concept of gradient discretisation and list the properties of the spaces and mappings that are important for the convergence analysis of the GDM. This convergence analysis is performed in Chapter 3 for linear and non-linear elliptic problems, in Chapter 5 for linear and non-degenerate non-linear parabolic problems, and in Chapter 6 for some degenerate parabolic problems. The idea of the GDM is then to mimick the weak formulation or the problem to be solved, by building, thanks to the function and gradient reconstructions, a discrete weak formulation which will be the gradient scheme. The convergence analysis of the GDM depends, of course, on the nature of the PDE to be solved. The definition the GDM, on the other hand, only depends to a large extent only on the boundary conditions. Therefore, the construction of a GD varies accordingly to the nature of these conditions. For simplicity, we start with homegeneous and non-homogeneous Dirichlet boundary conditions in Section 2.1, and then address the case of the home-

33 22 2 The gradient discretisation method geneous and non-homogeneous Neumann boundary conditions in Section 2.2. The gradient discretisations defined for these cases may then be easily generalised to the case of Robin conditions (Section 2.3) and mixed conditions (Section 2.4). Throughout this book, is a connected open bounded subset of R d which is the physical domain over which the p.d.e. is studied, d N is the space dimension, and p (1, + ) denotes a regularity index of the sought solution. In some abstract theorems, p might be allowed to take the value Dirichlet boundary conditions Homogeneous Dirichlet conditions Definition 2.1 (GD, homogeneous Dirichlet BCs). A gradient discretisation D for homogeneous Dirichlet conditions is defined by D = (X D,, Π D, D ), where: 1. the set of discrete unknowns X D, is a finite dimensional real vector space, 2. the function reconstruction Π D : X D, L p () is a linear mapping that reconstructs, from an element of X D,, a function over, 3. the gradient reconstruction D : X D, L p () d is a linear mapping which reconstructs, from an element of X D,, a gradient (vector-valued function) over. This gradient reconstruction must be chosen such that D := D L p () is a norm on X D,. d In the following chapters, we shall construct some gradient schemes for several problems, starting from a gradient discretisation. In order to show the convergence of the scheme, we use some properties of consistency and stability. As in the framework of the Finite Element method, stability is obtained through some uniform coercivity of the discrete operator which relies on a discrete Poincaré inequality. Definition 2.2 (Coercivity, Dirichlet BCs) If D is a gradient discretisation in the sense of Definition 2.1, define C D as the norm of the linear mapping Π D : C D = Π D v L max p (). (2.1) v X D, \{} v D A sequence ( ) m N of gradient discretisations in the sense of Definition 2.1 is coercive if there exists C P R + such that C Dm C P for all m N.

34 2.1 Dirichlet boundary conditions 23 Remark 2.3 (Discrete Poincaré inequality). Equation (2.1) yields the discrete Poincaré inequality Π D v L p () C D D v L p () for all v X D,. d The consistency properties that we need indicate how a regular function (and its gradient) are more or less well approximated by some function and gradient which are reconstructed from the space X D,. The function S D which we introduce hereafter is often called interpolation error in the framework of finite elements. Definition 2.4 (GD-consistency, homogeneous Dirichlet BCs) If D is a gradient discretisation in the sense of Definition 2.1, define S D : W 1,p () [, + ) by ϕ W 1,p (), S D (ϕ) = ( ) min Π D v ϕ v X L p () + Dv ϕ L p (). d D, (2.2) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.1 is GD-consistent, or consistent for short, if ϕ W 1,p (), lim m S (ϕ) =. (2.3) Remark 2.5 (Definition of the interpolant P D ). Since the L p () and L p () d norms are strictly convex, for each ϕ W 1,p () there is a unique P D ϕ X D, that realises the minimum in S D (ϕ), that is, such that We will write S D (ϕ) = Π D P D ϕ ϕ L p () + DP D ϕ ϕ L p () d. ( ) P D ϕ = argmin Π D v ϕ Lp () + Dv ϕ Lp (). d v X D, Note that P D, even though uniquely defined, is not necessarily a linear map. In the case p = 2, a linear interpolant P (2) D : W 1,p () X D, can be defined by setting ( ) 1/2 P (2) D ϕ = argmin Π D v ϕ 2 L 2 () + Dv ϕ 2 L 2 (). d v X D, This interpolant will be used to establish error estimates for linear parabolic equations (see the proof of Theorem 5.3, which also contains a proof of the linearity of P (2) D and of its approximation properties).

35 24 2 The gradient discretisation method A wellknown property of the gradient operator in H 1 is the so-called grad-div duality; the Stokes formula gives : ( u ϕ + u divϕ)dx =, u H 1 (), ϕ H div (). (2.4) where H div () = {ϕ L 2 () d : divϕ L 2 ()}. When dealing with nonconforming method, this property is no longer exact at the discrete level. The concept of limit-conformity which we now introduce states that the discrete gradient and divergence operator satisfy this property asymptotically. Since we shall be dealing with non linear problems, we introduce, or any q (1, + ), the space W q div () of functions in (Lq ()) d with divergence in L q (): W q div () = {ϕ Lq () d : divϕ L q ()}. (2.5) We recall that the space W 1,2 () is commonly denoted by H 1 () and that W 2 div () = H div(). Definition 2.6 (Limit-conformity, Dirichlet BCs) If D is a gradient discretisation in the sense of Definition 2.1, let p = p p 1 and define W D: W p div () [, + ) by ϕ W p div (), ( D u(x) ϕ(x) + Π D u(x)divϕ(x)) dx W D (ϕ) = sup. u X D, \{} u D (2.6) A sequence ( ) m N of gradient discretisations is limit-conforming if ϕ W p div (), lim m W (ϕ) =. (2.7) It is clear from its definition that the quantity W easures how well the reconstructed function and gradients satisfy the divergence (Stokes) formula (2.4). If the method is conforming in the sense that X D, is a subspace of W 1,p (), D u = u and Π D u = u for all u X D, W 1,p, the set {Π D v, v X D, } is a subspace of W 1,p () and for all v X D,, there exists u W 1,p () such that Π D v = u and D v = u. then W D. In general, W easures the defect of conformity of the method, and must vanish in the limit hence the name limit-conformity for the above property. The following equivalent condition for the limit-conformity property facilitates the proof of the regularity of a possible limit (Lemma 2.12 below). Lemma 2.7 (On limit-conformity, Dirichlet BCs). Let D be a gradient discretisation in the sense of Definition 2.1. Set p = p p 1 and define W D :

36 2.1 Dirichlet boundary conditions 25 W p div () X D, [, + ) by (ϕ, u) W p div () X D,, W D (ϕ, u) = ( D u(x) ϕ(x) + Π D u(x)divϕ(x)) dx. (2.8) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.1 is limit-conforming if and only if, for any sequence u m X Dm, such that ( u m Dm ) m N is bounded, Proof. ϕ W p div (), lim W Dm (ϕ, u m ) =. (2.9) m Let us remark that W D (ϕ) = sup u XD, \{} W D (ϕ, u) / u D. The proof that (2.7) implies (2.9) is then straightforward, since W D (ϕ, u) u D W D (ϕ). Let us prove the converse by way of contradiction. If (2.7) does not hold then there exists ϕ W p div (), ε > and a subsequence of () m N, still denoted by ( ) m N, such that W Dm (ϕ) ε for all m N. We can then find u m X Dm, \ {} such that W D (ϕ, u m ) 1 2 ε u m Dm. Considering the bounded sequence (u m / u m Dm ) m N, we get a contradiction with (2.9). Dealing with generic non-linearity often requires additional compactness properties on the scheme. Definition 2.8 (Compactness, Dirichlet BCs) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.1 is compact if, for any sequence u m X Dm, such that ( u m Dm ) m N is bounded, the sequence (Π Dm u m ) m N is relatively compact in L p (). Compactness is stronger than coercivity, as stated in the following lemma; in fact, coercivity is required in linear problems, whereas compactness is not (see Corollary 3.5 and Remark 3.6). Lemma 2.9 (Compactness implies coercivity). Let ( ) m N be a compact sequence of gradient discretisations in the sense of Definition 2.8. Then it is coercive in the sense of Definition 2.2. Proof. Let us assume that the sequence is not coercive. Then there exists a subsequence of ( ) m N (denoted in the same way) such that, for all m N, there exists u m X Dm, \ {} with

37 26 2 The gradient discretisation method Π Dm u m Lp () lim = +. m u m Dm Setting v m = u m / u m Dm, this gives lim m Π Dm v m Lp () = +. But v m Dm = 1 and the compactness of the sequence of discretisations therefore implies that the sequence (Π Dm v m ) m N is relatively compact in L p (). This gives a contradiction. Let us turn to a property that we shall often require on the function reconstruction Π D. Indeed, it is very often handy to obtain piecewise constant functions as approximate functions, the reason being that piecewise constant functions commute with any non-linearity. This will be a key argument for non linear problems. Definition 2.1 (Piecewise constant reconstruction) Let D = (X D,, Π D, D ) be a gradient discretisation in the sense of Definition 2.1. The operator Π D : X D, L p () is a piecewise constant reconstruction if there exists a basis (e i ) i B of X D, and a family of disjoint subsets ( i ) i B of such that Π D u = i B u i1 i for all u = i B u ie i X D,, where 1 i is the characteristic function of i. In other words, Π D u is the piecewise constant function equal to u i on i, for all i B. The set B is usually the natural set of (geometrical entities attached to the) degrees of freedom of the scheme. Moreover, Π D L p () is not requested to be a norm on X D,. Indeed, all degrees of freedom are involved in the definition of the reconstructed gradients, but in several examples some degrees of freedom are not used to reconstruct the functions itself. Hence some of the subsets i may be empty, which prevents Π D L p () from being a norm. Remark If Π D is a piecewise constant reconstruction and g : R R we have g(π D u(x)) = Π D g(u)(x) for a.e. x, u X D, where, for u = i B u ie i, we set g(u) = i B g(u) ie i X D, with g(u) i = g(u i ). We also have Π D u(x)π D v(x) = Π D (uv)(x) for a.e. x, u, v X D,, where uv X D, is defined by (uv) i = u i v i for all i B. Note that these definitions of g(u) or uv depend on the choice of the degrees of freedom I in X D,. We should therefore denote g B (u) or (uv) B to emphasize the dependency on B but, in practice, we will remove this exponent B as the degrees of freedom are usually canonically chosen and fixed throughout the whole study of a gradient scheme.

38 2.1 Dirichlet boundary conditions 27 The convergence analysis of sequences of approximate solutions to a partial differential equation (PDE) usually starts by finding a priori estimates on the solutions to the schemes. In the framework of gradient schemes, this means proving that u D D remains bounded. Lemma 2.12 below states that, if such a bound holds, we can find a weak limit to the reconstructed functions and their gradients. Combined if necessary with the compactness of the gradient discretisations, this opens the way to the last stage of the convergence analysis, which consists in showing that this limit is a solution to the PDE. Lemma 2.12 (Regularity of the limit, homogeneous Dirichlet BCs). Let ( ) m N be a sequence of gradient discretisations in the sense of Definition 2.1 which is coercive (Definition 2.2) and limit-conforming (Definition 2.6). Let u m X Dm, be such that ( u m Dm ) m N remains bounded. Then there exist a subsequence of (, u m ) m N, denoted in the same way, and u W 1,p () such that Π Dm u m converges weakly in L p () to u and Dm u m converges weakly in L p () d to u. Proof. Thanks to the coercivity, the sequence (Π Dm u m ) m N remains bounded in L p (). Therefore, there exists a subsequence of (, u m ) m N, denoted in the same way, and there exist u L p () and v L p () d such that Π Dm u m converges weakly in L p () to u and Dm u m converges weakly in L p () d to v. Extend Π Dm u m, u, Dm u m and v by outside ; the previous convergence results hold respectively in L p (R d ) and L p (R d ) d. Using the limit-conformity of ( ) m N and the bound on u m Dm, passing to the limit in (2.9) gives ϕ W p div (Rd ), (v(x) ϕ(x) + u(x)divϕ(x)) dx =. R d Being valid for any ϕ Cc (R d ) d, this relation proves both that v = u and that u W 1,p (). Let us now present some equivalent or sufficient conditions for the spaceconsistency, limit-conformity and compactness of a sequence of gradient discretisations. Lemma 2.13 (On GD-consistency, homogeneous Dirichlet BCs). A sequence ( ) m N of gradient discretisations is GD-consistent in the sense of Definition 2.4 if and only if there exists a dense subset R in W 1,p () such that ψ R, lim S (ψ) =. (2.1) m Proof. Let us assume that (2.1) holds and let us prove (2.3) (the converse is straightforward, take R = W 1,p ()). Let ϕ W 1,p () and ε >. Take ψ R such that ϕ ψ W 1,p () ε. For v X D,, the triangle inequality yields Π D v ϕ L p () + Dv ϕ L p () d

39 28 2 The gradient discretisation method Π D v ψ L p () + Dv ψ L p () d + ϕ ψ L p () + ϕ ψ L p () d. Hence, S Dm (ϕ) S Dm (ψ)+ ϕ ψ W 1,p () S (ψ)+ε, we get from (2.1) that lim sup m S Dm (ϕ) ε. The proof is then completed by letting ε. Lemma 2.14 (Equivalent condition for limit-conformity, Dirichlet BCs). Let ( ) m N be a coercive sequence of gradient discretisations in the sense of Definition 2.2. Then ( ) m N is limit conforming in the sense of Definition 2.6 if and only if there exists a dense subset S in W p div () (endowed with the norm ϕ W p = ϕ div () L p () + divϕ d L p ()) such that ψ S, lim W (ψ) =. (2.11) m Remark If is a locally star-shaped open set in R d (which is in particular the case if is polyhedral as in Section 7.1), then S = Cc (R d ) d is dense in W p div () and can therefore be used in Lemma Proof. Let C P R + be such that C Dm C P. Let us assume that (2.11) holds and let us prove (2.7). Let ϕ W p div (). Take ε > and ψ S such that ϕ ψ W p ε, which means that ϕ ψ div () L p () ε and d divϕ divψ L p () ε. We have, using the coercivity assumption, W Dm (ϕ) W Dm (ψ) + ϕ ψ L p () d + C P divϕ divψ L p () W Dm (ψ) + (1 + C P )ε. Using (2.11) we deduce that lim sup m W Dm (ϕ) (1 + C P )ε and the proof is completed by letting ε. Lemma 2.16 (Equivalent condition for compactness, Dirichlet BCs). Let D be a gradient discretisation in the sense of Definition 2.1 and let T D : R d R + be defined by ξ R d Π D v( + ξ) Π D v Lp (R, T D (ξ) = max d ), (2.12) v X D, \{} v D where Π D v has been extended by outside. The sequence ( ) m N is compact in the sense of Definition 2.8 if and only if lim sup T Dm (ξ) =. ξ m N

40 2.1 Dirichlet boundary conditions 29 Proof. This lemma is a consequence of Kolmogorov s compactness theorem in Lebesgue spaces. Step 1: we prove that the compactness of ( ) m N in the sense of Definition 2.8 is equivalent to the relative compactness in L p () of the set A = m N A m, where A m = Π Dm ({u X Dm,, u Dm = 1}). Indeed, any sequence in A is either contained in a finite union of A m, which means that it remains bounded in a finite dimensional space, or has a subsequence which can be written Π Dm(k) u m(k) for some increasing sequence (m(k)) k N N and some u m(k) X Dm(k), with u m(k) Dm(k) = 1. Hence, the compactness of ( ) m N gives the relative compactness of A in L p (). Moreover, any sequence u m X Dm, such that u m Dm is bounded can be written u m = λ m u m with (λ m ) m N bounded and u m Dm = 1. We have then Π Dm u m = λ m v m for some v m A and the relative compactness of A in L p () therefore shows that ( ) m N is compact in the sense of Definition 2.8. Step 2: a statement of Kolmogorov s theorem. Let w L p (R d ) be the extension of w L p () by outside. A classical statement of Kolmogorov s compactness theorem is: A is relatively compact in L p () if and only it is bounded in L p () and if τ A (ξ) := sup w( + ξ) w L p (R d ) as ξ. w A But τ A is sub-additive. Indeed, for all ξ, ξ R d we have w( + ξ + ξ ) w L p (R d ) w( + ξ + ξ ) w( + ξ ) L p (R d ) + w( + ξ ) w L p (R d ) = w( + ξ) w L p (R d ) + w( + ξ ) w L p (R d ), and therefore τ A (ξ + ξ ) τ A (ξ) + τ A (ξ ). Hence, if lim ξ τ A (ξ) =, then τ A is finite on a neighborhood of in R d and its sub-additivity shows that it is in fact finite on R d. Now, taking ξ R d such that ξ > diam(), for all w A we see that w( + ξ) and w have disjoint supports and therefore ( ) 1/p τ A (ξ ) w(x + ξ ) p dx + w(x) p dx = 2 1/p w Lp(). R d R d The finiteness of τ A (ξ ) then ensures that A is bounded in L p (). Kolmogorov s theorem can therefore be re-stated as: A is relatively compact in L p () if and only if lim ξ τ A (ξ) =. Step 3: conclusion. We have

41 3 2 The gradient discretisation method A = ({ }) v Π Dm, v X Dm, \ {} v Dm m N and thus, since T Dm (ξ) = max v X Dm, \{} ( v Π v Dm ) ( + ξ) Π Dm ( v v Dm ) L p (R d ) (the functions being extended by outside ), we deduce sup m N T Dm (ξ) = τ A (ξ). The conclusion then follows from Steps 1 and 2. The following lemma is an immediate consequence of the previous ones and facilitates, in many practical situations, the verification of the core properties of gradient discretisations. Lemma 2.17 (Sufficient conditions, homogeneous Dirichlet BCs). Let F be a family of gradient discretisations in the sense of Definition 2.1. Assume that there exist C, ν (, ) and, for all D F, a real value h D (, + ) such that: S D (ϕ) Ch D ϕ W 2, (), for all ϕ C c (), (2.13a) W D (ϕ) Ch D ϕ (W 1, (R d )) d, for all ϕ C c (R d ) d, (2.13b) Π D v( + ξ) Π D v L max p (R d ) C ξ ν, for all ξ R d. (2.13c) v X D, \{} v D Then any sequence ( ) m N F such that h Dm as m is GDconsistent, limit-conforming and compact (and therefore coercive). Note that for several gradient schemes, the parameter h D in the above lemma can simply be chosen as the mesh size Non-homogeneous Dirichlet boundary conditions We present here the framework of gradient discretisations for diffusion problems with non-homogeneous Dirichlet boundary conditions. To handle nonhomogeneous boundary conditions, we need the concept of trace of functions in W 1,p (), for p (1, + ). This concept requires more regularity on than in Section and we therefore assume here that has a Lipschitz boundary. We recall in Section some facts about the trace operator γ. Definition 2.18 (GD, non-homogeneous Dirichlet BCs). A gradient discretisation D for non-homogeneous Dirichlet conditions is defined by D = (X D, I D,, Π D, D ) where: 1. the set of discrete unknowns X D = X D, X D, is the direct sum of two finite dimensional spaces on R, corresponding respectively to the interior degrees of freedom and to the boundary degrees of freedom,

42 2.1 Dirichlet boundary conditions the linear mapping I D, : W 1 1 p,p ( ) X D, is an interpolation operator for the traces γu of the elements u W 1,p (), 3. the function reconstruction Π D : X D L p () is linear, 4. the gradient reconstruction D : X D L p () d is linear. It must be chosen such that D := D Lp () is a norm on X D,. d Remark 2.19 (Domain of I D, ). The interpolation operator I D, does not necessarily need to be defined on the whole space W 1 1 p,p ( ). If g is the boundary condition of the considered problem (e.g. in (3.22b)), we only need to define I D, g. Hence, if g has a better regularity than W 1 1 p,p ( ), we can take advantage of this to find a simpler definition of I D, g, see for example Remark In that case, the GD-consistency (Definition 2.2) is required only for functions ϕ W 1,p () such that γϕ has the additional regularity supposed when constructing I D,. Coercivity, limit-conformity, compactness and piecewise constant reconstructions are defined as in the homogeneous case, by considering Definitions 2.2, 2.6, 2.8 and 2.1 on the spaces X D,. The definition of GDconsistency needs to be modified and implicitly imposes assumptions on the interpolation operator. Definition 2.2 (GD-consistency, non-homogeneous Dirichlet BCs) If D is a gradient discretisation in the sense of Definition 2.18, define S D : W 1,p () [, + ) by ϕ W 1,p (), S D (ϕ) = min{ Π D v ϕ L p () + D v ϕ Lp () d, v I D, γϕ X D, }. (2.14) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.18 is GD-consistent if ϕ W 1,p (), lim m S (ϕ) =. (2.15) Since coercivity, limit-conformity and compactness are the same as for homogeneous Dirichlet conditions, the characterisation Lemmas 2.7, 2.14 and 2.16 may also be used in the context of non-homogeneous Dirichlet conditions. It will be useful, as in the homogeneous case, to also have a characterisation of the GD-consistency using dense subsets of W 1,p (). This characterisation however requires an additional assumption on the trace interpolation operator, stating that for any given trace on, we can find elements in X D which interpolate this trace and have a norm controled by this trace.

43 32 2 The gradient discretisation method Lemma 2.21 (Equivalent condition for GD-consistency, non-homogeneous Dirichlet BCs). Let ( ) m N be a sequence of gradient discretisations in the sense of Definition We assume that there exists C 1 such that, for any m N and any ϕ W 1,p (), min{ Π Dm v Lp () + Dm v L p () d, v I, γϕ X D, } C 1 ϕ W 1,p (). (2.16) Then ( ) m N is GD-consistent in the sense of Definition 2.2 if and only if there exists a dense subset R in W 1,p () such that ψ R, lim S (ψ) =. (2.17) m Remark Note that (2.16) is almost a requirement to GD-consistency in the sense of Definition 2.2. Indeed, for any ϕ W 1,p (), taking and element in X D + I D, γϕ X D, which realises the minimum S D (ϕ), we see that min{ Π D v L p () + D v Lp () d, v I D, γϕ X D, } S D (ϕ) + ϕ W 1,p (). Hence, if ( ) m N is GD-consistent in the sense of Definition 2.2, estimate (2.16) is asymptotically true as m since S Dm (ϕ). Proof. The proof is very similar as the proof of Lemma 2.13 and we obviously only have to prove the if direction (the only if holds with R = W 1,p ()). Let ϕ W 1,p () and ε >. Take ψ R such that ϕ ψ W 1,p () ε. Let v X Dm, + I Dm, γψ which realises the minimum in S Dm (ψ) and let w X Dm, + I Dm, γ(ϕ ψ) which realises the minimum for ϕ ψ in the left-hand side of (2.16). Then v + w X Dm, + I Dm, γϕ and, therefore, S Dm (ϕ) Π Dm (v + w) ϕ Lp () + Dm (v + w) ϕ L p () d Π Dm v ψ Lp () + Dm v ψ L p () d + Π Dm w Lp () + Dm w L p () d + ϕ ψ Lp () + ϕ ψ Lp () d S Dm (ψ) + (C 1 + 1) ϕ ψ W 1,p () S Dm (ψ) + (C 1 + 1)ɛ. The conclusion follows as in the proof of Lemma The convergence properties imposed on the interpolant I D, are somewhat hidden in the definition of GD-consistency. The following lemma shows that the formulation (3.26) of gradient schemes for non-homogeneous Dirichlet conditions make sense (for linear as well as non-linear problems): sequences of solutions to the gradient schemes indeed converge, up to a subsequence, to a function that has the required trace on the boundary of.

44 2.2 Neumann boundary conditions 33 Lemma 2.23 (Regularity of the limit, non-homogeneous Dirichlet BCs). Let ( ) m N be a sequence of gradient discretisations in the sense of Definition 2.18 which is coercive (Definition 2.2), limit-conforming (Definition 2.6) and GD-consistent (Definition 2.2). Let g W 1 1 p,p ( ). Let u m X Dm be such that u m I Dm, g X Dm, and ( Dm u m Lp () d) m N remains bounded. Then there exist a subsequence of (, u m ) m N, denoted in the same way, and u W 1,p () such that γu = g and, as m, Π Dm u m converges weakly in L p () to u and Dm u m converges weakly in L p () d to u. Proof. Let g W 1,p () such that γ g = g. By GD-consistency of ( ) m N, we can find v m X Dm, + I Dm, g such that Π Dm v m g in L p () and Dm v m g in L p () d. By assumption, u m v m = (u m I Dm, g) + (I Dm, g v m ) belongs to X Dm,, and Dm (u m v m ) L p () remains bounded. Hence, by recalling that the d coercivity and limit-conformity of ( ) m N are identical to the coercivity and limit-conformity of the underlying gradient discretisations for homogeneous Dirichlet conditions (i.e. with X Dm, instead of X Dm ), Lemma 2.12 shows that, up to a subsequence, Π Dm (u m v m ) ũ weakly in L p () and Dm (u m v m ) ũ weakly in L p () d, where ũ W 1,p (). The properties of (v m ) m N then show that Π Dm u m = Π Dm (u m v m ) + Π Dm v m ũ + g =: u in L p (), and Dm u m = Dm (u m v m ) + Dm v m ũ + g = u in L p () d. The function u = ũ + g belongs to W 1,p () and has trace γu = γũ + γ g = + g = g. 2.2 Neumann boundary conditions Homogeneous Neumann conditions We take here a connected open bounded subset of R d with Lipschitz boundary and p (1, + ). Definition 2.24 (GD, homogeneous Neumann BCs). A gradient discretisation D for homogeneous Neumann conditions is defined by D = (X D, Π D, D ) where: 1. the set of discrete unknowns X D is a finite dimensional vector space on R, 2. the function reconstruction Π D : X D L p () is linear, 3. the gradient reconstruction D : X D L p () d is linear. The operators D and Π ust be chosen such that ( v D := D v p 1/p + L p () d Π D v(x)dx p) (2.18) is a norm on X D.

45 34 2 The gradient discretisation method Remark The choice of v D, involving the integral of Π D v rather that its L p () norm, is justified by the way the scheme for Neumann problem is written and the a priori estimates that can be established on the approximate solution, cf Section The discrete properties of gradient discretisations for Neumann problems, that ensure the convergence of the associated gradient schemes, are the following. Definition 2.26 (Coercivity, homogeneous Neumann BCs) If D is a gradient discretisation in the sense of Definition 2.24, define C D = Π D v L max p (). (2.19) v X D \{} v D A sequence ( ) m N of gradient discretisations in the sense of Definition 2.24 is coercive if there exists C P R + such that C Dm C P for all m N. Definition 2.27 (GD-consistency, Neumann BCs) If D is a gradient discretisation in the sense of Definition 2.24, define S D : W 1,p () [, + ) by ϕ W 1,p (), S D (ϕ) = min v X D ( ΠD v ϕ Lp () + D v ϕ Lp () d ). (2.2) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.24 is GD-consistent if ϕ W 1,p (), lim m S (ϕ) =. (2.21) Definition 2.28 BCs) (Limit-conformity, homogeneous Neumann For p (1, + ), let p = p p 1 and W p div, () = {ϕ Lp () d : divϕ L p (), γ n (ϕ) = }, where γ n (ϕ) is the normal trace of ϕ on (see Section 2.2.3). If D be a gradient discretisation in the sense of Definition 2.24, define W D : () [, + ) by W p div,

46 2.2 Neumann boundary conditions 35 ϕ W p div, (), W D (ϕ) = max v X D \{} 1 v D D v(x) ϕ(x)dx + Π D v(x)divϕ(x)dx. (2.22) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.24 is limit-conforming if ϕ W p div, (), lim W (ϕ) =. (2.23) m Definition 2.29 (Compactness, homogeneous Neumann BCs) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.24 is compact if, for any sequence u m X Dm such that ( u m Dm ) m N is bounded, the sequence (Π Dm u m ) m N is relatively compact in L p (). Note that the definition of piecewise constant reconstruction for a gradient discretisation for homogeneous Neumann boundary conditions is the same as Definition 2.1, replacing the space X D, by X D. As in the case of Dirichlet boundary conditions (see Lemma 2.13), the GDconsistency (resp. the limit-conformity) of sequences of gradient discretisations in the case of homogeneous Neumann conditions needs only be checked on a dense subset of W 1,p () (resp. W 1,p ()). The proof of this result which is stated in the following lemma is identical to the proof of Lemma 2.13, replacing W 1,p () by W 1,p (). Lemma 2.3 (Equivalent condition for GD-consistency, Neumann BCs). A sequence ( ) m N of gradient discretisations is GD-consistent in the sense of Definition 2.27 if and only if there exists a dense subset R in W 1,p () such that ϕ R, lim S (ϕ) =. m We stated in Lemma 2.16 a compactness criterion in the case of Dirichlet boundary conditions; we state below a similar criterion for the case of Neumann Boundary conditions, which holds under an additional regularity property on the domain. Contrary to the case of Dirichlet boundary conditions, we may no longer prolong the functions by outside of, and therefore the criterion can only involve the interior translations

47 36 2 The gradient discretisation method Lemma 2.31 (A criterion for compactness). Let be an open set of R d satisfying the segment condition : there exist open sets (U i ) i=1,...,k and nonzero vectors (ξ i ) i=1,...,k such that k i=1 U i and, for all i = 1,..., k and all t (, 1], U i + tξ i. Let p 1 be given and (u m ) m N be a bounded sequence in L p () such that lim ξ sup m N u m ( + ξ) u m L p ( ξ ) = (where ξ = {x, [x, x + ξ] }). (2.24) Then (u m ) m N is relatively compact in L p (). Proof. Let us first notice that, for any ω relatively compact in, we have ω ξ for ξ small enough. Hence, by the classical Kolmogorov compactness theorem, there exists u L p loc () and a subsequence, still denoted by (u m) m N, such that u m u in L p loc (). Since (u m) m N is bounded in L p (), Fatou s lemma shows that u belongs in fact to L p (). We infer that (u m u) m N is bounded in L p () and satisfies (2.24). Reasoning on u m u rather than u, we can therefore assume that u = and we have to prove that u m in L p (). The main issue is of course to estimate this convergence on a neighborhood of. Let (U i ) i=1,...,k and (ξ i ) i=1,...,k be given by the segment condition for. For any i {1,..., k} and any r (, 1], K i,r = U i + rξ i is a compact subset of. Moreover, for any m N, by the change of variable y = x + rξ i, we get u m (x) p dx u m (y rξ i ) p dy U i U i+rξ i 2 p 1 U i+rξ i u m (y rξ i ) u m (y) p dy +2 p 1 K i,r u m (y) p dy. For any z U i and any s [, 1] we have (z + rξ i ) srξ i = z + (1 s)rξ i, by definition of z if s = 1 and by definition of ξ i if s < 1. Hence, U i + rξ i rξi and the preceding inequality gives u m (x) p dx 2 p 1 η( rξ i ) + 2 p 1 u m (y) p dy U i K i,r where η(ξ) = sup m N u m ( + ξ) u m p L p ( ξ ) tends to as ξ. Summing all these inequalities on i = 1,..., k and defining the open set U = k i=1 U i, neighborhood of in R d, we obtain U k u m (x) p dx 2 p 1 η( rξ i ) + 2 p 1 i=1 k i=1 K i,r u m (y) p dy.

48 2.2 Neumann boundary conditions 37 Let us now take ε > and fix r (, 1] such that, for all i = 1,..., k, η( rξ i ) ε. Since, for any i, K i,r is a compact subset of we have K i,r u m (y) p dy as m and therefore lim sup u m (x) p dx 2 p 1 kε. m U The proof is completed by letting ε Non-homogeneous Neumann conditions We present here the framework of gradient discretisations for diffusion problems with non-homogeneous Neumann boundary conditions. We again take a connected open bounded subset of R d with Lipschitz boundary and p (1, + ). Definition 2.32 (GD, non-homogeneous Neumann BCs). A gradient discretisation D for non-homogeneous Neumann conditions D is defined by D = (X D, Π D, T D, D ) where: 1. the set of discrete unknowns X D is a finite dimensional vector space on R, 2. the function reconstruction Π D : X D L p () is linear, 3. the trace reconstruction T D : X D L p ( ) is linear; it provides, from an element of X D, a function over, 4. the gradient reconstruction D : X D L p () d is linear. The operators D and Π ust be chosen such that is a norm on X D. ( v D := D v p + L p () d 1/p Π D v(x)dx p) (2.25) The discrete properties of gradient discretisations for Neumann problems, that ensures the the convergence of the associated gradient schemes, are the following. The GD-consistency and piecewise constant reconstruction are still defined by Definitions 2.27 and 2.1 (replacing X D, with X D in the latter definition). Definition 2.33 (Coercivity, non-homogeneous Neumann BCs) If D is a gradient discretisation in the sense of Definition 2.32, define

49 38 2 The gradient discretisation method C D = ( { ΠD v Lp () max max, T }) Dv Lp ( ). (2.26) v X D \{} v D v D A sequence ( ) m N of gradient discretisations in the sense of Definition 2.32 is coercive if there exists C P R + such that C Dm C P for all m N. (Limit-conformity, non-homogeneous Neu- Definition 2.34 mann BCs) For p (1, + ), let p = p p 1 and W p div, () = {ϕ Lp () d : divϕ L p (), γ n (ϕ) L p ( )}, (2.27) where γ n (ϕ) is the normal trace of ϕ on (see Section 2.2.3). If D is a gradient discretisation in the sense of Definition 2.32, define W D : W p div, () [, + ) by ϕ W p div, (), 1 W D (ϕ) = max v X D \{} v D D v(x) ϕ(x)dx + Π D v(x)divϕ(x)dx T D v(x)γ n (ϕ)(x)ds(x). (2.28) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.32 is limit-conforming if ϕ W p div, (), lim W (ϕ) =. (2.29) m Remark This definition of limit-conformity ensure both that the dual operator to Dm approximates the continuous divergence operator, and that T Dm approximates the continuous trace operator (see also Lemma 2.4). Definition 2.36 BCs) (Compactness, non-homogeneous Neumann A sequence ( ) m N of gradient discretisations in the sense of Definition 2.32 is compact if, for any sequence u m X Dm such that ( u m Dm ) m N is bounded, the sequence (Π Dm u m ) m N is relatively

50 2.2 Neumann boundary conditions 39 compact in L p () and the sequence (T Dm u m ) m N is weakly relatively compact in L p ( ). Note that the weak relative compactness of the sequence of discrete traces is an immediate consequence of the coercivity of the sequence of gradient discretisations. Moreover, as in the case of Dirichlet boundary conditions, the compactness of a sequence of gradient discretisations implies its coercivity. Remark As for Dirichlet problems, compactness of the gradient discretisation is only useful to deal with non-linearities in the equation. If these non-linearity involve the trace of the solution, then the compactness property should be modified and include also the relative strong compactness of (T Dm u m ) m N in L p ( ). Lemma 2.38 (Equivalent condition for limit-conformity, non-homogeneous Neumann BCs). Let ( ) be a sequence of coercive sequence of gradient discretisations in the sense of Definition Then ( ) m N is limit-conforming in the sense of Definition 2.34 if and only if there exists a dense subset S in W p div, () (endowed with the norm ϕ = W p div, () ϕ L p () + divϕ d L p () + γ n(ϕ) L p ( )) such that ϕ S, lim m W (ϕ) =. (2.3) Remark Lemma 2.46 shows that the set S = C (R d ) d W div,p, (). is dense in Proof. Let C P R + be such that C Dm C P. Assuming that (2.3) holds, we take ϕ W p div, () and, for ε >, select ϕ S such that ϕ ϕ ε. W p div, () Then, using the definition of C P, W Dm (ϕ) W Dm (ϕ) + ϕ ϕ Lp () d +( divϕ divϕ L p () + γ n (U) γ n (ϕ) L p ( ) )C P W Dm (ϕ) + (1 + 2C P )ε. From (2.3) with ϕ instead of ϕ we then deduce lim sup m W Dm (ϕ) (1 + 2C P )ε and the proof is complete. The following lemma is the equivalent of Lemma 2.12 for non-homogeneous Neumann conditions. Lemma 2.4 (Regularity of the limit, non-homogeneous Neumann BCs). Let ( ) m N be a coercive and limit-conforming sequence of gradient discretisations in the sense of Definitions 2.33 and Let u m X Dm be

51 4 2 The gradient discretisation method such that ( u m Dm ) m N is bounded. Then there exists u W 1,p () such that, up to a subsequence, Π Dm u m u weakly in L p (), (2.31) T Dm u m γu weakly in L p ( ), (2.32) Dm u m u weakly in L p () d. (2.33) Remark This lemma shows in particular that if u m X Dm is such that, for some u W 1,p (), Π Dm u m u weakly in L p () and Dm u m u weakly in L p () d, then T Dm u m γu weakly in L p ( ). Remark In the case of gradient discretisations for homogeneous Neumann conditions, which do not involve a trace reconstruction, Lemma 2.4 is still valid with (2.32) removed. Proof. By coercivity, the bound on u m Dm shows that Π Dm u m Lp (), T Dm u m L p ( ) and u m L p () d are bounded. There exists therefore u L p (), w L p ( ) and v L p () d such that, up to a subsequence, Π Dm u m u weakly in L p (), T Dm u m w weakly in L p ( ) and Dm u m v weakly in L p () d. Let ϕ W p div, (). By Definition (2.28) of W D, u m Dm W Dm (ϕ) ( Dm u m (x) ϕ(x) + Π Dm u(x)divϕ(x))dx T Dm u m (x)γ n (ϕ)(x)ds(x). (2.34) By limit-conformity of ( ) m N and the bound on u m Dm, the left-hand side of this expression tends to as m. Hence, passing to the limit we get (v(x) ϕ(x) + u(x)divϕ(x))dx w(x)γ n (ϕ)(x)ds(x) =. (2.35) Applied to ϕ Cc () d, this relation shows that v = u and therefore that u W 1,p () and that (2.31) and (2.33) hold. For any l L p () (W 1 1 p,p ( )), take ϕ W p div () such that γ n(ϕ) = l (see Lemma 2.45). Then ϕ W p div, () and therefore, (2.35) and the definition (2.39) of γ n (ϕ) = l (with u instead of L γu, see the comments after (2.39)) give l(x)γu(x)ds(x) = w(x)l(x)ds(x), which proves that w = γu. Hence (2.32) is verified and the proof is complete.

52 2.2 Neumann boundary conditions 41 We complete this section by stating an approximation property of T D. This property is useful to deduce error estimates on the traces of gradient scheme approximations to linear elliptic problems (see Remark 3.12). Proposition 2.43 (Approximation property of T D Neumann BCs). Let D be a gradient discretisation in the sense of Definition We define, for ϕ W 1,p (), S D (ϕ) = min w X D ( Π D w ϕ Lp () + T Dw γϕ Lp ( ) + D w ϕ L p () d ). (2.36) Then, for any v X D and any ϕ W 1,p (), ) T D v γϕ Lp ( ) C D ( 1 p Π D v ϕ + Lp() Dv ϕ Lp() ) d + max (1, C D, C D 1 p S D (ϕ). Remark The quantity S D in (2.36) is actually the measure of the GDconsistency for Fourier boundary conditions (see (2.49)). Proof. We introduce ( P D ϕ = argmin Π D w ϕ L p () + T Dw γϕ L p ( ) w X D and we notice that + D w ϕ L p () d ) Π D P D ϕ ϕ L p () + T DP D ϕ γϕ L p ( ) + D P D ϕ ϕ Lp () d S D(ϕ). (2.37) By definition of C D and of D, Hölder s inequality gives, for all w X D, ( ) T D w Lp ( ) C D D w Lp () + 1 d p Π D w Lp (). (2.38) A triangular inequality therefore provides T D v γϕ Lp ( ) T D (v P D ϕ) Lp ( ) + T DP D ϕ γϕ Lp ( ) C D ( 1 p Π D v Π D P D ϕ Lp () + Dv D P D ϕ Lp () d ) + T D P D ϕ γϕ L p ( ). We then use the triangular inequality again in the first two terms in the righthand side, to introduce ϕ in the first one and ϕ in the second one, and we apply (2.37) to conclude the proof.

53 42 2 The gradient discretisation method Complements on trace operators Let be a bounded open subset of R d with Lipschitz boundary. The trace operator γ : W 1,p () W 1 1 p,p ( ) is well defined and surjective, and there exists a linear continuous lifting operator L : W 1 1 p,p ( ) W 1,p () such that γl g = g for any g W 1 1 p,p ( ). We recall that in the case p = 2, W 1 2,2 ( ) is generally denoted by H 1 2 ( ) and the set W 1,2 () is denoted H 1 (). We can then define the normal trace γ n (ϕ) (W 1 1 p,p ( )) of ϕ W p div () (where p = p p 1 ) the following way. Denoting by, the duality product between (W 1 1 p,p ( )) and W 1 1 p,p ( ), we let, for any g W 1 1 p,p ( ), γ n (ϕ), g = (ϕ(x) L g(x) + L g(x)divϕ(x)) dx. (2.39) The linearity and continuity of L ensure that γ n (ϕ) is indeed an element of (W 1 1 p,p ( )). Moreover, for any ϕ W 1,p () such that γϕ = g we have L g ϕ W 1,p () and thus, by Stokes formula, ( (L g ϕ)(x) ϕ(x) + (L g ϕ)(x)divϕ(x)) dx =. This shows that (2.39) is also valid if we replace L g with any ϕ W 1,p () having trace g on. Lemma 2.45 (Surjectivity of the normal trace). The normal trace γ n : W p div () (W 1 1 p,p ( )) is linear continuous surjective. More precisely, there exists C > depending only on and p such that, for any l (W 1 1 p,p ( )), there exists ϕ W p div () satisfying γ n(ϕ) = l and ϕ W p C l div () 1 1 (W p,p ( )), (2.4) where we recall that ϕ W p div () = ϕ L p () d + divϕ L p (). Proof. Since γ : W 1,p () W 1 1 p,p ( ), for any l (W 1 1 p,p ( )) we have γ l (W 1,p ()). There exists thus (h, ϕ) L p () L p () d such that, for all ϕ W 1,p (), and l, γϕ = γ l, ϕ (W 1,p ()),W 1,p () = (ϕ(x) ϕ(x) + h(x)ϕ(x)) dx (2.41) ϕ L p () d + h L p () γ l (W 1,p ()) C l (W 1 1 p,p ( )) (2.42)

54 2.2 Neumann boundary conditions 43 where C is the norm of γ (it is also the norm of γ ). Testing (2.41) with functions ϕ in C c () shows that h = divϕ and, therefore, that ϕ W p div (). Taking then a generic g W 1 1 p,p ( ) and applying (2.41) with ϕ = L g gives γ n (ϕ) = l and Estimate (2.42) gives (2.4). Lemma 2.46 (Density of smooth functions in W div,p, ()). Let be a polytopal open set (see Definition 7.2) and let p (1, + ). The space W div,p, () is defined by (2.27) and endowed with the norm ϕ W div,p, () = ϕ Lp () d + divϕ L p () + γ n(ϕ) L p ( ). Then 1. C c () d is dense in {ϕ W div,p, () : γ n (ϕ) = }, 2. C c (R d ) d is dense in W div,p, (). Remark We only stated the lemma for polytopal open sets, but the proof shows that the result is more general than this (in particular, it holds for open sets with piecewise C 1,1 boundary since the normal n is then Lipschitz continuous outside a set of zero (d 1)-dimensional measure). Proof. Item 1: using the localisation techniques of [7, Ch. 1, Theorem 1.1, (iii)], we can reduce the study to the case where is strictly star-shaped, say with respect to. This means that, for any λ (, 1), λ. Let ϕ W div,p, () such that γ n (ϕ) =. Then the extension ϕ of ϕ to R d by outside belongs to W p div (Rd ), since the normal traces of ϕ is continuous through. Let λ (, 1) and define ϕ λ : x ϕ(x/λ). As λ 1, we have ϕ λ ϕ in L p () d and div( ϕ λ ) = λ 1 (div ϕ)( /λ) div ϕ in L p (). Moreover, the support of ϕ λ is contained in λ, and is therefore compact in. Let (ρ ɛ ) ɛ> be a smoothing kernel. For ɛ small enough, ϕ λ ρ ɛ belongs to Cc () d. As ɛ, we also have ϕ λ ρ ɛ ϕ λ in L p () d and div( ϕ λ ρ ɛ ) = div( ϕ λ ) ρ ɛ div( ϕ λ ) in L p (). Hence, letting in that order λ 1 and ɛ, the functions ( ϕ λ ρ ɛ ) give an approximation in W div,p, () of ϕ by functions in Cc () d, and the proof of Item 1 is complete. Item 2: let ϕ W div,p, () and ε >. In the following, C denotes a generic constant, independent on ε but whose value can change from one occurrence to the other. Since is polytopal, n is piecewise constant and thus smooth outside a set S of measure in. We can therefore find a function ψ ε that is C -smooth, vanishes on a neighborhood of S, and such that γ n (ϕ) ψ ε L p ( ) ε. Since ψ ε vanishes on a neighbourhood of the singularities S of n, we can find a function ψ ε Cc (R d ) d such that γ n (ψ ε ) = ψ ε (simply extend, on a neighbourhood U of, the smooth function ψ ε n into a function that does not depend on the coordinate orthogonal to n, and multiply this extension by a function in Cc (U) equal to 1 on a neighbourhood of ).

55 44 2 The gradient discretisation method Let us consider the function ϕ ψ ε W div,p, (). We have γ n (ϕ ψ ε ) L p ( ) = γ n(ϕ) ψ ε L p ( ) ε. (2.43) By Lemma 2.45 (applied with p and p swapped) and since L p ( ) is embedded in (W 1 1 p,p ( )), we can find ζ ε W p div () such that and γ n (ζ ε ) = γ n (ϕ ψ ε ) (2.44) ζ ε W p div () C γ n(ϕ ψ ε ) L p ( ) Cε. (2.45) Property (2.44) shows that ζ ε W p (2.45), that div, () and, combined with (2.43) and ζ ε W div,p, () = ζ ε W p div () + γ n(ζ ε ) L p ( ) Cε. (2.46) The function ϕ ψ ε ζ ε therefore belongs to W p div, () and satisfies γ n(ϕ ψ ε ζ ε ) =. By Item 1 we can find ξ ε Cc () d such that (ϕ ψ ε ζ ε ) ξ ε W div,p, () ε. (2.47) We now set ϕ ε = ψ ε + ξ ε. This function belongs to C c (R d ) d and ϕ ϕ ε = (ϕ ψ ε ζ ε ξ ε ) + ζ ε so, by (2.46) and (2.47), ϕ ϕ ε W div,p, () Cε. 2.3 Non-homogeneous Fourier boundary conditions Although we shall present few convergence results for PDE s with Fourier (also known as Robin) boundary conditions, for the sake of completeness we give here the gradient discretisation framework for these conditions. Here is again a connected open bounded subset of R d with Lipschitz boundary and p (1, + ). Except for the choice of the norm D, the definition of a gradient discretisation for Fourier boundary conditions is the same as for Neumann boundary conditions. Definition 2.48 (GD, non-homogeneous Fourier BCs). A gradient discretisation D for non-homogeneous Fourier conditions D is defined by D = (X D, Π D, T D, D ) where: 1. the set of discrete unknowns X D is a finite dimensional vector space on R,

56 2.3 Non-homogeneous Fourier boundary conditions the function reconstruction Π D : X D L p () is linear, 3. the trace reconstruction T D : X D L p ( ) is linear, 4. the gradient reconstruction D : X D L p () d is linear. The operators D and T ust be chosen such that ( 1/p v D := D v p + T L p () d D v p L ( )) (2.48) p is a norm on X D. The coercivity, limit-conformity, compactness and piecewise constant reconstruction for gradient discretisations for non-homogeneous Fourier conditions are defined exactly as for non-homogeneous Neumann conditions, i.e. Definitions 2.33, 2.34, 2.36 and 2.1 (with X D, replaced by X D in the latter, and using the norm (2.48) in all these definitions). The GD-consistency, however, must take into account the trace reconstruction. Definition 2.49 BCs) (GD-consistency, non-homogeneous Fourier If D is a gradient discretisation in the sense of Definition 2.48, define S D : W 1,p () [, + ) by ϕ W 1,p ()(, S D (ϕ) = min ΠD v ϕ Lp () + T D v γϕ Lp ( ) v X D ) + D v ϕ Lp (). d (2.49) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.48 is GD-consistent if ϕ W 1,p (), lim m S (ϕ) =. (2.5) We notice that Lemma 2.3 (characterisation of GD-consistency using a dense set of W 1,p ()), Lemma 2.38 (characterisation of limit-conformity using a dense set of W p div, ()) and Lemma 2.4 (regularity of the limit) still hold in the framework of gradient schemes for non-homogeneous Fourier boundary conditions. For Fourier boundary conditions, the reconstructed trace has been included in the definition of S D, and we can therefore expect an approximation property as in Proposition However, the norm is different and actually already includes the reconstructed trace. For this reason, an additional assumption must be introduced which states that the reconstructed trace can be controlled by the reconstructed function and gradients, see (2.51). In practice, for many gradient discretisation this assumption is easy to check by using Lemma B.16 and the notion of control by a polytopal toolbox (cf. Section 7.2.3).

57 46 2 The gradient discretisation method The proof of this proposition is identical to the proof of Proposition 2.43, the assumption (2.51) playing the role of (2.38). Proposition 2.5 (Approximation property of T D Fourier BCs). Let D be a gradient discretisation in the sense of Definition We assume that there exists θ > such that ( ) v X D : T D v Lp ( ) θ Π D v Lp () + Dv Lp (). (2.51) d Then, for any v X D and any ϕ W 1,p (), T D v γϕ Lp ( ) θ ( Π D v ϕ Lp () + Dv ϕ Lp () d ) + max(1, θ)s D (ϕ). 2.4 Mixed boundary conditions From the framework of non-homogeneous Dirichlet and Neumann boundary conditions, it is very easy to construct a gradient scheme discretisation for mixed boundary conditions. We consider here p (1, ), a connected open bounded subset of R d with Lipschitz boundary and we assume that Γ d, Γ n are two disjoint relatively open subsets of such that \(Γ d Γ n ) = and Γ d > (2.52) ( denotes here the (d 1)-dimensional measure). Definition 2.51 (GD, mixed BCs). Under Assumption (2.52), a gradient discretisation D for mixed boundary conditions D is defined by D = (X D, I D,Γd, Π D, T D,Γn, D ) where: 1. the set of discrete unknowns X D = X D,,Γn X D,Γd is the direct sum of two finite dimensional vector spaces on R, corresponding respectively to the degrees of freedom in and on Γ n and to the degrees of freedom on Γ d, 2. the linear mapping I D,Γd : W 1 1 p,p ( ) X D,Γd is an interpolation operator for the restrictions (γu) Γd of traces of elements u W 1,p (), 3. the function reconstruction Π D : X D L p () is linear, 4. the trace reconstruction T D,Γn : X D L p (Γ n ) is linear, and reconstructs from an element of X D a function over Γ n, 5. the gradient reconstruction D : X D L p () d is linear. The operator ust be chosen such that is a norm on X D,,Γn. v D := D v Lp () d

58 2.4 Mixed boundary conditions 47 Definition 2.52 (Coercivity, mixed BCs) Under Assumption (2.52), if D is a gradient discretisation in the sense of Definition 2.51, define ( { ΠD v L C D = max max p (), T }) D,Γ n v Lp (Γ n). v X D,,Γn \{} v D v D (2.53) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.51 is coercive if there exists C P R + such that C Dm C P for all m N. Definition 2.53 (GD-consistency, mixed BCs) Under Assumption (2.52), if D is a gradient discretisation in the sense of Definition 2.51, define S D : W 1,p () [, + ) by ϕ W 1,p (), S D (ϕ) = min{ Π D v ϕ L p ()+ D v ϕ Lp () d, v I D,Γd γϕ X D,,Γn }. (2.54) A sequence ( ) m N of gradient discretisations in the sense of Definition 2.51 is GD-consistent if ϕ W 1,p (), lim m S (ϕ) =. (2.55) Definition 2.54 (Limit-conformity, mixed BCs) For p (1, + ), let p = p p 1 and W div,p,γ n () = {ϕ L p () d : divϕ L p (), γ n (ϕ) L p (Γ n )}, (2.56) where γ n (ϕ) is the normal trace of ϕ on. Under Assumption (2.52), if D is a gradient discretisation in the sense of Definition 2.51, define W D : W div,p,γ n () [, + ) by ϕ W div,p,γ n (), { 1 W D (ϕ) = max v D ( D v(x) ϕ(x) + Π D v(x)divϕ(x)) dx } T D,Γn v(x)γ n (ϕ)(x)ds(x), v X D,,Γ n \ {}. (2.57) Γ n

59 48 2 The gradient discretisation method A sequence ( ) m N of gradient discretisations in the sense of Definition 2.51 is limit-conforming if ϕ W div,p,γ n (), lim W (ϕ) =. (2.58) m Remark Note that γ n (ϕ) L p (Γ n ) makes sense because Γ n is a relatively open subset of. Indeed, when ϕ L p () d and divϕ L p () then γ n (ϕ) (W 1 1 p,p ( )) and saying that this linear form belongs to L p (Γ n ) means by definition that there exists g L p (Γ n ) such that, for any w W 1 1 p,p ( ) with support in Γ n, γ n (ϕ), w (W 1 1 p,p ( )),W 1 1 p,p ( ) = Γ n g(x)w(x)ds(x). Definition 2.56 (Compactness, mixed BCs) Under Assumption (2.52), a sequence ( ) m N of gradient discretisations in the sense of Definition 2.51 is said to be compact if, for any sequence u m X Dm such that ( u m Dm ) m N is bounded, the sequence (Π Dm u m ) m N is relatively compact in L p () and the sequence (T Dm,Γ n u m ) m N is weakly relatively compact in L p (Γ n ). The definition of piecewise constant reconstruction for a gradient discretisation for mixed boundary conditions is the same as Definition 2.1, replacing the space X D, by X D. As in the non-homogeneous Neumann case, the weak compactness of the reconstructed traces is an immediate consequence of the coercivity of the sequence of gradient discretisations. The equivalent of Lemmas 2.12, 2.23 and 2.4 is the following lemma. Lemma 2.57 (Regularity of the limit, mixed BCs). Under Assumption (2.52), let ( ) m N be a coercive, GD-consistent and limit-conforming sequence of gradient discretisations in the sense of Definitions 2.52, 2.53 and Let g W 1 1 p,p ( ) and, for m N, let u m X Dm such that u m I Dm,Γ d g X D,,Γn and ( Dm u m L p () d) m N is bounded. Then, there exists u W 1,p () such that γu = g on Γ d and, up to a subsequence, as m, Π Dm u m u weakly in L p (), (2.59) Dm u m u weakly in L p () d. If we assume moreover that g =, or that there exists ϕ g W 1,p () such that γϕ g = g and, as m,

60 2.4 Mixed boundary conditions 49 min{ Π Dm v ϕ g L p () + T Dm,Γ n v γϕ g L p (Γ n) then we also have + Dm v ϕ g L p () d, v I,Γ d γϕ X Dm,,Γ n }, (2.6) T Dm,Γ n u m (γu) Γn weakly in L p (Γ n ). (2.61) Remark Assumption (2.6), if satisfied for all ϕ g W 1,p (), is similar to the GD-consistency of ( ) m N in the sense of Fourier boundary conditions (but with a trace reconstruction only on Γ n ). Proof. Step 1: we suppose that g =. Since u m = u m I Dm,Γ d g X Dm,,Γ n, we proceed with u m as in the proof of Lemma 2.4, replacing W p div, () with W div,p,γ n () and, in (2.34), the integrals over with integrals over Γ n. This gives u L p (), v L p () d and w L p (Γ n ) such that, up to a subsequence, Π Dm u m u weakly in L p (), Dm u m v weakly in L p () d, T Dm,Γ n u m w weakly in L p (Γ n ), and such that the following version of (2.35) holds for all ϕ W div,p,γ n (): (v(x) ϕ(x) + u(x)divϕ(x))dx w(x)γ n (ϕ)(x)ds(x) =. Γ n Selecting ϕ Cc () d gives v = u, and therefore u W 1,p (). Taking then ϕ smooth that does not vanish on and using an integration-by-parts, we obtain γ n (ϕ)(x)γu(x)ds(x) = w(x)γ n (ϕ)(x)ds(x). Γ n This shows that γu = w on Γ n and that γu = on Γ d, which concludes the proof of (2.59) and (2.61) if g =. Step 2: we consider a general g W 1 1 p,p ( ). As in the proof of Lemma 2.23, we take an extension g W 1,p () of g and we use the GD-consistency to find v m X Dm such that v m I Dm,Γ d g X Dm,,Γ n, Π Dm v m g in L p () and Dm v m g in L p () d. Then u m v m X Dm,,Γ n and we can apply the reasoning in Step 1 to this function. We therefore find U W 1,p () such that γu = on Γ d and, up to a subsequence, Π Dm (u m v m ) U weakly in L p (), Dm (u m v m ) U weakly in L p () d, T Dm,Γ n (u m v m ) γu weakly in L p (Γ n ). (2.62) We then let u = U + g W 1,p (), so that γu = g on Γ d. The convergence properties of (v m ) m N and (2.62) then show that (2.59) holds. Step 3: we consider a general g W 1 1 p,p ( ), and we assume that (2.6) holds.

61 5 2 The gradient discretisation method Then we can take v m X Dm such that v m I Dm,Γ d g X Dm,,Γ n, Π Dm v m g in L p (), T Dm,Γ n v m γ g = g in L p (Γ n ), and Dm v m g in L p () d. We can then reproduce Step 2 with this v m. Since T Dm,Γ n v m g in L p (), the convergence T Dm,Γ n (u m v m ) γu = γu g in L p (Γ n )-weak (see (2.62)) ensures that (2.61) holds.

62 3 Elliptic problems The ingredients necessary to implement the gradient discretisation method (GDM) for a diffusion problem were introduced in Chapter 2 for various boundary conditions (BCs). They can now be applied to write a gradient scheme (GS) for elliptic and parabolic PDEs; The present chapter is devoted to the study of gradient schemes for the approximation of linear and nonlinear elliptic problems. Error estimates are provided in the linear case. In the non-linear case, the convergence is proved thanks to compactness arguments. 3.1 The linear case In this section, linear problems are considered, so that p = 2 is chosen in all the definitions of Chapter Homogeneous Dirichlet boundary conditions We consider here the following problem: div(λ(x) u) = f + div(f ) in, with boundary conditions u = on, under the following assumptions: is an open bounded connected subset of R d (d N ), Λ is a measurable function from to the set of d d symmetric matrices and there exists λ, λ > such that, for a.e. x, Λ(x) has eigenvalues in [λ, λ], f L 2 (), F L 2 () d. (3.1a) (3.1b) (3.2a) (3.2b) (3.2c)

63 52 3 Elliptic problems Note that the assumptions on the right hand side include both the case of a right hand side in L 2 () (taking F = ) and the case of a right hand side in H 1 (), since H 1 () = {divv, v L 2 () d }. Note also that the symmetry assumption on Λ(x) is not mandatory to study the convergence of the GDM for (3.1a), but it is commonly satisfied in applications. Under these hypotheses, the weak solution of (3.1) is the unique function u satisfying: u H 1 (), v H 1 (), Λ(x) u(x) v(x)dx = f(x)v(x)dx F (x) v(x)dx. Let us now introduce an approximation of Problem (3.3) by the GDM. (3.3) Definition 3.1 (GS, homogeneous Dirichlet BCs). If D = (X D,, Π D, D ) is a GD in the sense of Definition 2.1, then the related gradient scheme for Problem (3.3) is defined by Find u X D, such that for any v X D,, Λ(x) D u(x) D v(x)dx = f(x)π D v(x)dx F (x) D v(x)dx. (3.4) Note that, considering a basis (ξ (i) ) i=1,...,n of the space X D,, Scheme (3.4) is equivalent to solving the linear square system AU = B, letting u = N U j ξ (j), j=1 A ij = B i = Λ(x) D ξ (j) (x) D ξ (i) (x)dx, (3.5) f(x)π D ξ (i) (x)dx F (x) D ξ (i) (x)dx. The following theorem, first proved in [52] in the case F =, is in the spirit of the second Strang lemma [69]. Theorem 3.2 (Control of the approximation error, homogeneous Dirichlet BCs). Under Assumptions (3.2), let u H 1 () be the solution of Problem (3.3) (which implies that in the distribution sense div(λ u + F ) = f L 2 () and therefore Λ u + F H div ()). Let D be a GD in the sense of Definition 2.1. Then there exists one and only one u D X D, solution to the GS (3.4); this solution satisfies the following inequalities: u D u D L 2 () 1 [ d WD (Λ u + F ) + (λ + λ)s D (u) ], (3.6) λ

64 u Π D u D L2 () 1 λ 3.1 The linear case 53 [ CD W D (Λ u + F ) + (C D λ + λ)s D (u) ], (3.7) where C D, S D and W D are respectively the norm of the reconstruction operator Π D, the space-consistency defect and the conformity defect, defined by (2.1) (2.6). Remark 3.3 (Mesh-based GSs). Gradient schemes are often mesh-based. Under usual non-degeneracy assumptions on the meshes, it is proved in many cases that there exists C R +, not depending on D, such that and ϕ H 2 () H 1 (), S D (ϕ) Ch D ϕ H2 () ϕ H 1 () d, W D (ϕ) Ch D ϕ H 1 () d, where h easures the mesh size. The proofs of these inequalities are done, for some important examples of gradient schemes, in Part III. Then, under the assumptions that the coefficients of Λ are Lipschitz-continuous, that F H 1 () and that u H 2 () H 1 (), Theorem 3.2 gives in fact O(h D ) error estimates. Remark 3.4 (Super-convergence) As noticed in Remark 3.3 above, the L 2 estimate in Theorem 3.2 only provides an O(h D) rate of convergence for low-order schemes. It is well known that several of these schemes, e.g. conforming and non-conforming P 1 finite elements, enjoy a higher rate of convergence in L 2 norm. This phenomenon is known as super-convergence. It is possible to establish, in the framework of gradient schemes, an improved L 2 estimate that provides such super-convergence results for various schemes, including some for which super-convergence was previously not proved. See [42]. Proof. Let us first prove that, if (3.6) holds for any solution u D X D, to Scheme (3.4), then the solution to Scheme (3.4) exists and is unique. Indeed, let us prove that, assuming (3.6), the matrix denoted by A of the linear system (3.5) is non-singular. This will be completed if we prove AU = implies U =. Thus, we consider the particular case where f = and F = which gives a zero right-hand side. In this case the solution u of (3.3) is equal to zero a.e. Then from (3.6), we get that any solution to the scheme satisfies u D D =. Since D is a norm on X D, this leads to u D =. Therefore (3.5) (as well as (3.4)) has a unique solution for any right-hand side f and F. Let us now prove that any solution u D X D, to Scheme (3.4) satisfies (3.6) and (3.7). As noticed in the statement of the theorem, we can take ϕ = Λ u + F H div () in the definition (2.6) of W D. We then obtain, for a given v X D,, D v(x) (Λ(x) u(x) + F (x)) + Π D v(x)div(λ u + F )(x)dx

65 54 3 Elliptic problems which leads, since f = div(λ u + F ) a.e., to D v(x) (Λ(x) u(x) + F (x)) Π D v(x)f(x)dx v D W D (Λ u + F ), v D W D (Λ u + F ). (3.8) Since u D is a solution to (3.4), we get Λ D v(x) ( u(x) D u D (x))dx v D W D (Λ u + F ). (3.9) We define P D u = argmin ( Π D w u L 2 () + D w u L2 () d) (3.1) w X D, and we notice that, by definition (2.2) of S D, Π D u u L 2 () + DP D u u L 2 () d = S D(u). (3.11) Recalling the definition of D in Definition 2.1, by (3.9) we get Λ D v(x) ( D P D u(x) D u D (x))dx D v L 2 () W D(Λ u + F ) d + Λ(x) D v(x) ( D P D u(x) u(x))dx (3.12) ( ) D v L 2 () W d D (Λ u + F ) + λ D P D u u L 2 () d D v L 2 () d ( WD (Λ u + F ) + λs D (u) ). Choosing v = P D u u D yields λ D (P D u u D ) L 2 () d W D(Λ u + F ) + λs D (u) (3.13) and (3.6) follows by writing u D u D L2 () d u D P D u L 2 () + D(P d D u u D ) L 2 () d S D (u) + 1 ( WD (Λ u + F ) + λs D (u) ). λ (3.14) Using (2.1) and (3.13), we get λ Π D P D u Π D u D L 2 () C D (W D (Λ u + F ) + λs D (u)),

66 3.1 The linear case 55 which yields (3.7) by using, as in (3.14), a triangular inequality and the estimate u Π D P D u L2 () S D(u). d Theorem 3.2 gives a control of the approximation error thanks to the spaceconsistency indicator S D, the limit-conformity indicator W D and the coercivity indicator C D. This theorem yields the convergence of the GDM for sequences of GDs that are space-consistent, limit-conforming and coercive, as stated in Corollary 3.5 below. Can this be re-stated as the usual Consistency and Stability = Convergence (3.15) statement, well-known in the context of finite difference schemes? The answer to this question is yes, provided a correct definition of consistency is chosen. In the classical finite difference setting, the consistency error measures (roughly speaking) how well the exact solution fits into the scheme. Formally, assume that the equation to be discretised is written under the form Lu = f, and that the scheme is under the form L h u h = f h = Π h f where h denotes the discretisation step and, for a given function g, Π h g is the vector whose components are the values (g(x i )) i=1,...,n of g at the discretisation points (x i ) i=1,...,n. Then the consistency error for the finite difference scheme is defined by c h = L h Π h u f h = L h Π h u Π h (Lu). In this context, it is well-known that (3.15) holds: indeed, consistency (i.e. c h as h ) and stability (i.e. L 1 h bounded) imply convergence (i.e. max i=1,...,n Π h (u) u h (x i ) as h ). In the finite element context, (3.15) no longer holds under these terms, although the spirit remains the same. The reason for the failure of (3.15) is that consistency no longer refers to how the exact solution fits into the complete scheme, but only into the discrete equation of the scheme. To be more explicit, consider the following elliptic problem: u V, (3.16) a(u, v) = (f, v), v V, (3.17) where V = H 1 (), f L 2 (), and a is a continuous coercive bilinar form on V. Consider a finite element scheme for the discretisation of Problem (3.17), which reads u h V h, (3.18) a h (u h, v) = (f, v), v V h, (3.19) where V h is a finite dimensional space. In order to measure how well the exact solution fits into the scheme, the consistency error should measure (i) how far V h is from V,

67 56 3 Elliptic problems (ii) how far κ h is from, with κ h = a h (Π h u, v) (f, v) max, (3.2) v V h \{} v V where Π h u is either u itself, or some kind of interpolant of u. In most finite element textbooks, these two notions have been separated: Property (i) is measured by the so-called interpolation error, while the term consistency (or asymptotic consistency) only refers to the fact that κ h = (or κ h as h ). We shall call this latter property FEM-consistency for the sake of clarity. For the conforming P 1 finite element for instance, a h = a, Π h u can be taken equal to u, and κ h =, in which case the finite element scheme is said to be consistent. However, there are cases where the solution to the PDE itself cannot be plugged into the scheme s equation (for instance when using numerical quadrature), but when an interpolant of this solution needs to be used; for more on this, see e.g. [21, Chapter 4] or [46, Chapter 2]. Hence in the FEM context, (3.15) still holds provided Consistency = FEM-consistency and interpolation error control. (3.21) For the stability issue in the FEM context, we also refer to [46]. Let us now view a finite element method as a GDM. In this context, the space-consistency (see Definition 2.4) together with the limit-conformity (see Definition 2.2) is sufficient to ensure the consistency of the scheme in sense (3.21). Indeed, in the context of the GDM, the equivalent of the term κ h defined by (3.2) reads (for F = ) κ D = Λ(x) D (P D u)(x) D v(x)dx Λ(x) u(x) D v(x)dx } D v L 2 () {{ d } Controlled by S D(ū), see (3.12) Λ(x) u(x) D v(x)dx f(x)π D v(x)dx + } D v L 2 () {{ d } Controlled by W D(λ ū), see (3.8) Note that space-consistency and stability (or coercivity) are not sufficient to prove the convergence of a general GDM. The limit-conformity (which is inherent to all conforming finite element methods) is needed to ensure that the discrete function reconstruction and the discrete gradient reconstruction are chosen in a coherent way. Hence for the GDM, we may also write provided Consistency and Stability = Convergence,.

68 3.1 The linear case 57 Consistency = Space-consistency and Limit-conformity. Let us conclude this section by stating the convergence of the GS, which follows easily from Theorem 3.2. Corollary 3.5 (Convergence). Under Hypotheses (3.2), let ( ) m N be a sequence of GDs in the sense of Definition 2.1, which is coercive, spaceconsistent and limit-conforming in the sense of definitions 2.2, 2.4 and 2.6. Then, for any m N, there exists a unique solution u m X Dm, to the gradient scheme (3.4) and Π Dm u m converges in L 2 () to the solution u of (3.3) and Dm u m converges in L 2 () d to u as m. Remark 3.6 (On the compactness assumption). Note that, in the present linear case, the compactness of the sequence of GDs is not required to obtain the convergence ; this compactness assumption will be needed in general for non linear problems (see Remark 3.36 for non linear problems that do not require it). Remark 3.7 (Space-consistency and limit conformity are necessary conditions) We state here a kind of reciprocal property to Corollary 3.5. Let us assume that, under Assumptions (3.2a)-(3.2b), a sequence () m N of GDs is such that, for all f L 2 () and F L 2 () d and for all m N, there exists u m X Dm, which is solution to the gradient scheme (3.4) and which satisfies that Π Dm u m (resp. Dm u m) converges in L 2 () to the solution u of (3.3) (resp. in L 2 () d to u). Then () m N is space-consistent and limit-conforming in the sense of definitions 2.4 and 2.6. Indeed, for ϕ H 1 (), let us consider f = and F = Λ ϕ in (3.3). Since in this case, u = ϕ, the assumption that Π Dm u m (resp. Dm u m) converges in L 2 () to the solution ϕ of (3.3) (resp. converges in L 2 () d to ϕ) suffices to prove that S Dm (ϕ) tends to as m, and therefore the sequence () m N is space-consistent. For ϕ H div (), let us set f = divϕ and F = ϕ in (3.3). In this case, the solution u is equal to a.e., since the right-hand side of (3.3) vanishes for any v H 1 (). Since u m X Dm, is a solution to the GS (3.4), we get for all v X Dm,, ( Dm v(x) ϕ(x) + Π Dm v(x)divϕ(x)) dx Using that Dm u m converges in L 2 () d to, we get that λ Dm u m L 2 () d Dm v L 2 () d. W Dm (ϕ) λ Dm u m L 2 () d as m, hence concluding that the sequence () m N is limit-conforming. Note that, if we now assume that Dm u m converges only weakly (instead of strongly) in L 2 () d to u, the same conclusion holds. Indeed, these hypotheses are sufficient to prove that Dm u m converges in L 2 () d to u. It suffices to observe that lim (f(x)π Dm u m m(x) F (x) Dm u m(x))dx

69 58 3 Elliptic problems = (f(x)u(x) F (x) u(x))dx. Then we take v = u in (3.3) and v = u m in (3.4), this leads to Λ(x) Dm u m(x) Dm u m(x)dx = Λ(x) u(x) u(x)dx. lim m In addition to the assumed weak convergence property of Dm u m, this proves Λ(x)( Dm u m(x) u(x)) ( Dm u m(x) u(x))dx =, lim m and the convergence of Dm u m to u in L 2 () d follows Non-homogeneous Dirichlet boundary conditions As already mentioned in Section 2.1.2, the case of non-homogeneous Dirichlet boundary conditions requires the concept of trace of functions in H 1 () which demands the Lipschitz regularity of the boundary. We refer to Section for the properties of the trace operator in this context. We consider the linear problem defined in its strong form by: div(λ(x) u) = f + div(f ) in, (3.22a) with boundary conditions under Assumptions (3.2) and u = g on, (3.22b) is an open bounded connected subset of R d (d N ) with Lipschitz boundary, (3.23) g H 1/2 ( ). (3.24) Under these hypotheses, the weak solution of (3.22a) is the unique function u satisfying: u {w H 1 (), γ(w) = g}, v H 1 (), Λ(x) u(x) v(x)dx = f(x)v(x)dx F (x) v(x)dx. (3.25) Under Assumptions (3.2)-(3.24), the GDM applied to Problem (3.25) yields the following gradient scheme. Definition 3.8 (GS, non-homogeneous Dirichlet BCs). If D = (X D = X D, X D,, I D,, Π D, D ) is a gradient discretisation in the sense of Definition 2.18, then we define the related gradient scheme for (3.25) by

70 3.1 The linear case 59 Find u I D, g + X D, such that, for any v X D,, Λ(x) D u(x) D v(x)dx = f(x)π D v(x)dx F (x) D v(x)dx. (3.26) The following theorem states error estimates for the GS for non-homogeneous Dirichlet boundary conditions. This theorem yields a convergence result (not explicitly stated) similar to Corollary 3.5. Theorem 3.9 (Control of the approximation error, non-hom. Dirichlet BCs). Under Hypotheses (3.2), (3.23), (3.24), let u H 1 () be the solution of (3.25) (remark that since f L 2 (), one has Λ u + F H div ()). Let D be a GD in the sense of Definition Then there exists one and only one u D X D solution to the GS (3.26), and it satisfies the following inequalities: u D u D L2 () 1 [ d WD (Λ u + F ) + (λ + λ)s D (u) ], (3.27) λ u Π D u D L 2 () 1 λ [ CD W D (Λ u + F ) + (C D λ + λ)s D (u) ], (3.28) where C D, S D and W D are defined by Definitions 2.2, 2.2 and 2.6. Proof. Reasoning as in the proof of Theorem 3.2, we arrive at (3.9) for any v X D,. We then define P D u = argmin w I D, g+x D, ( Π D w u L 2 () + D w u L2 () d), and we notice that, by definition (2.14) of S D, (3.11) is still valid. Moreover, the vector v = P D u u D belongs to I D, g + X D, + ( I D, g + X D, ) = X D, and can therefore be used in (3.9). The rest of the proof is then exactly as in the proof of Theorem Neumann boundary conditions We consider here a linear elliptic problem with non-homogeneous Neumann boundary conditions div(λ(x) u) = f + div(f ) in, Λ u n + F n = h on, (3.29) where n is the unit normal outward to, assumed to be Lipschitz, under Assumptions (3.2), (3.23) and h L 2 ( ) and f(x)dx + h(x)ds(x) =. (3.3)

71 6 3 Elliptic problems Under these hypotheses and defining H 1 () = {ϕ H 1 (), ϕ(x)dx = }, the weak formulation of (3.29) is u H (), 1 v H (), 1 Λ(x) u(x) v(x)dx = f(x)v(x)dx F (x) v(x)dx + h(x)γ(v)(x)ds(x). (3.31) We recall that, owing to Hypothesis (3.3), Problem (3.31) is equivalent to u H 1 (), v H 1 (), Λ(x) u(x) v(x)dx + u(x)dx v(x)dx = f(x)v(x)dx F (x) v(x)dx + h(x)γ(v)(x)ds(x), (3.32) since letting v 1 in (3.32) implies that u(x)dx =. The approximation of Problem (3.31) by the GDM is described in the next definition. Definition 3.1 (GS, Neumann BCs). If D = (X D, Π D, T D, D ) is a GD for Neumann problems in the sense of Definition 2.32, then we define the related gradient scheme for (3.31) by Find u X D such that, for anyv X D, Λ(x) D u(x) D v(x)dx + Π D u(x)dx = f(x)π D v(x)dx F (x) D v(x)dx + h(x)t D v(x)ds(x). Π D v(x)dx (3.33) The error estimates for Neumann boundary conditions are stated in Theorem We do not explicitly the convergence result, similar to Corollary 3.5, that stems from these error estimates. Theorem 3.11 (Control of the approximation error, Neumann BCs). Under Hypotheses (3.2), (3.23) and (3.3), let u H 1 () be the solution of (3.31) (remark that f L 2 () and h L 2 ( ) imply that in Λ u + F W div,2, (), see (2.27)). Let D be a GD for a Neumann problem in the sense of Definition Then there exists one and only one u D X D solution to the GS (3.33), and this element satisfies the following inequalities:

72 3.1 The linear case 61 u D u D L2 () d Err + S D(u), (3.34) u Π D u D L2 () C D Err + S D (u), (3.35) 1 [ ] with Err := W D (Λ u + F ) + (λ + 1/2 C D )S D (u), min(λ, 1) where C D, S D and W D are defined by (2.26), (2.2) and (2.28). Remark 3.12 (Error estimate on the traces). If we let S D be the measure of space-consistency for Fourier boundary conditions (i.e. (2.36)), then Proposition 2.43 and Theorem 3.11 show that γ(u) T D u D L 2 ( ) C 1 ( WD (Λ u + F ) + S D (u) ), where C 1 only depends on λ, λ, and an upper bound of C D. Proof. Recall that in Definition 2.32, it is assumed that v D := ( D v 2L2()d + Π D v(x)dx 2 ) 1/2 defines a norm on X D. Therefore, proving (3.34) for any solution u D X D to Scheme (3.33) is enough to prove the existence and uniqueness of this solution (because this estimate shows that, whenever f =, h = and F =, the only possible solution to the scheme is u D = ). To prove the estimates, we take ϕ = Λ u + F W div,2, () in the definition (2.28) of W D and using that u is the solution to (3.31). We then have, for any v X D, [ D v(x) (Λ(x) u(x) + F (x)) Π D v(x)f(x) ] dx h(x)t D v(x)ds(x) v D W D (Λ u + F ). Therefore, since u D is a solution to (3.33), we get Λ D v(x) ( u(x) D u D (x))dx Π D u D (x)dx Π D v(x)dx v D W D (Λ u + F ). We then introduce P D u = argmin ( Π D w u L 2 () + D w u L 2 () d) (3.36) w X D in the above inequality. It leads to

73 62 3 Elliptic problems Λ D v(x) ( D P D u(x) D u D (x))dx + (Π D P D u(x) Π D u D (x))dx Π D v(x)dx v D W D (Λ u + F ) + Λ D v(x) ( D P D u(x) u(x))dx + Π D P D u(x)dx Π D v(x)dx. Observing that Π D P D u(x)dx = we can write, using the definition (2.26) of C D, Λ D v(x) ( D P D u(x) D u D (x))dx (Π D P D u(x) u(x))dx 1/2 Π D P D u u L2 (), + (Π D P D u(x) Π D u D (x))dx Π D v(x)dx ] v D [W D (Λ u + F ) + (λ + 1/2 C D )S D (u). We now let v = P D u u D. Recalling the definition (2.25) of D, we obtain P D u u D D W D(Λ u + F ) + (λ + 1/2 C D )S D (u). min(λ, 1) The conclusion follows as in the proof of Theorem Mixed boundary conditions To conclude the case of linear problem, we consider here a linear elliptic problem with mixed boundary conditions div(λ(x) u) = f + div(f ) in, u = g on Γ d, Λ u n + F n = h on Γ n, (3.37) under Assumptions (3.2), (3.23), (2.52) and

74 3.2 Unknown-dependent diffusion problems 63 g H 1/2 ( ), h L 2 (Γ n ). (3.38) Denoting by H 1 Γ d () the set of functions in H 1 () whose trace on Γ d vanishes, the weak formulation of (3.37) is u {w H 1 () : γ(w) = g on Γ d }, v HΓ 1 d (), Λ(x) u(x) v(x)dx = f(x)v(x)dx F (x) v(x)dx + h(x)γ(v)(x)ds(x). Γ n The GDM applied to this mixed problem yields the following scheme. (3.39) Definition 3.13 (GS, mixed linear problem). If D = (X D, I D,Γd, Π D, T D,Γn, D ) is a GD for mixed problems in the sense of Definition 2.51, then the related gradient scheme for (3.39) is defined by: Find u I D,Γd g + X D,,Γn such that, for any v X D,,Γn, Λ(x) D u(x) D v(x)dx = f(x)π D v(x)dx F (x) D v(x)dx + h(x)t D,Γn v(x)ds(x). Γ n (3.4) The proof of the following error estimates for mixed boundary conditions is similar to the proofs made in the case of other boundary conditions. Likewise, a convergence result similar to Corollary 3.5 follows from these error estimates. Theorem 3.14 (Control of the approximation error). Under Hypotheses (3.2), (3.23), (2.52) and (3.38), let u H 1 () be the solution of (3.39) (remark that since f L 2 () and h L 2 (Γ n ), we have Λ u+f W div,2,γn (), see (2.56)). Let D be a GD for mixed boundary conditions in the sense of Definition Then there exists one and only one u D X D solution to the gradient scheme (3.4), and this element satisfies the following inequalities: u D u D L 2 () 1 [ d WD (Λ u + F ) + (λ + λ)s D (u) ], λ u Π D u D L 2 () 1 λ [ CD W D (Λ u + F ) + (C D λ + λ)s D (u) ], where C D, S D and W D are defined by (2.53), (2.54) and (2.57). 3.2 Unknown-dependent diffusion problems We shall consider here the quasi-linear operator 1 u div(λ(x, u(x)) u), which is often used to model non-linear heterogeneous materials. For such an 1 Recall that a partial differential operator is said to be quasilinear if it is linear with respect to all the highest order derivatives of the unknown function.

75 64 3 Elliptic problems operator, we remain in the functional framework of the linear case, considering again that p = Homogeneous Dirichlet boundary conditions We consider the following problem: div(λ(x, u(x)) u) = f in, with boundary conditions u = on, under the following assumptions: is an open bounded connected subset of R d, d N, Λ is a Caratheodory function from R to M d (R), Λ(x, s) is measurable w.r.t. x and continuous w.r.t. s, there exists λ, λ > such that, for a.e. x, for all s R Λ(x, s) is symmetric with eigenvalues in [λ, λ], f L 2 (). (3.41a) (3.41b) (3.42a) (3.42b) (3.42c) Under these hypotheses, a weak solution of (3.41a) is a function u (not necessarily unique) satisfying: u H 1 (), v H 1 (), Λ(x, u(x)) u(x) v(x)dx = f(x)v(x)dx. (3.43) Then Problem (3.43) is approximated under Assumptions (3.42) by the following gradient scheme. Definition 3.15 (Gradient scheme, unknown-dependent diffusion). If D = (X D,, Π D, D ) is a GD in the sense of Definition 2.1, then we define the related gradient scheme for (3.3) by Find u X D, such that, for any v X D,, Λ(x, Π D u(x)) D u(x) D v(x)dx = f(x)π D v(x)dx. (3.44) Note that, considering a basis (ξ (i) ) i=1,...,n of the space X D,, Scheme (3.44) is equivalent to solving the system of N non-linear equations with N unknowns A(u)U = B with u = N U j ξ (j), j=1

76 3.2 Unknown-dependent diffusion problems 65 A ij (u) = Λ(x, Π D u(x)) D ξ (j) (x) D ξ (i) (x)dx, (3.45) B i = f(x)π D ξ (i) (x)dx. Standard methods for the approximation of a solution of this system can be considered, such as the fixed point method A(u (k) )U (k+1) = B or the Newton- Raphson method. Let us now state a convergence result (note that an error estimate between an approximate solution and a weak solution to (3.44) cannot be stated, since the uniqueness of the solution to neither (3.45) nor (3.44) is known in the general case). Theorem 3.16 (Convergence, unknown-dependent diffusion). Under assumptions (3.42), take a sequence ( ) m N of GDs in the sense of Definition 2.1, which is space-consistent, limit-conforming and compact in the sense of Definitions 2.4, 2.6 and 2.8 (it is then coercive in the sense of Definition 2.2, see Lemma 2.9). Then, for any m N, there exists at least one u m X Dm, solution to the gradient scheme (3.44) and, up to a subsequence, Π Dm u m converges strongly in L 2 () to a solution u of (3.43) and Dm u m converges strongly in L 2 () d to u as m. In the case where the solution u of (3.43) is unique, then the whole sequence converges to u as m in the senses above. Proof. Step 1: existence of a solution to the scheme. Let D = (X D,, Π D, D ) be a GD in the sense of Definition 2.1. Let w X D, be given, and let u X D, be such that Find u X D, such that, v X D,, Λ(x, Π D w(x)) D u(x) D v(x)dx = f(x)π D v(x)dx. (3.46) Therefore, u is solution to the square linear system A(w)U = B, where A(w) and B are defined in (3.45). Let us prove that the matrix A(w) is invertible. Letting v = u in (3.46), and applying the Cauchy-Schwarz inequality and Hypothesis (3.42b), we get λ D u 2 L 2 () d f L 2 () Π D u L 2 () C D f L 2 () D u L2 () d, where C D is defined by (2.1) in Definition 2.2. This shows that D u L2 () d C D λ f L 2 (). (3.47) This completes the proof that A(w) is invertible, since (3.47) shows that A(w)U = implies U =. We then can define the mapping F : R N R N,

77 66 3 Elliptic problems by F (W ) = U, with U is the solution of the linear system A(w)U = B. This mapping is continuous, thanks to the continuity of the coefficients of the inverse of a matrix with respect to its coefficients. Moreover, we get from (3.47) that some norm of U remains bounded, which means that F maps R N into some closed ball B if R N. Therefore the Brouwer fixed point theorem (Theorem C.2) proves that the equation F (U) = U has at least one solution. This proves the existence of at least one discrete solution to (3.44). We note that the previous estimates easily show that any solution to this scheme satisfies (3.47). Step 2: convergence of Π Dm u m and Dm u m. Thanks to the coercivity hypothesis and (3.47), we have Dm u m L2 () d C P λ f L 2 (). (3.48) We may then apply Lemma 2.12, which states that there exists a subsequence of (, u m ) m N, denoted in the same way, and there exists u H 1 () such that Dm u m converges weakly in L 2 () d to u and Π Dm u m converges weakly in L 2 () to u. Thanks to the compactness hypothesis, there exists again a subsequence of the preceding one, denoted in the same way, such that Π Dm u m converges in L 2 () to u. Step 3: proof that u is a solution to Problem (3.43). This proof is done by passing to the limit in the gradient scheme (3.44), considering as test function the following interpolation of a given function ϕ H 1 (). Let us define, for a given GD D, P D : H 1 () X D, by ( ) P D ϕ = argmin Π D v ϕ L2 () + Dv ϕ L2 (). d v X D, We have Π D (P D ϕ) ϕ L 2 () + D(P D ϕ) ϕ L 2 () S D(ϕ) d and therefore, by space-consistency of the sequence ( ) m N, Π Dm (P Dm ϕ) ϕ strongly in L 2 () and Dm (P Dm ϕ) ϕ strongly in L 2 () d. Using Lemma C.4 page 46 (non-linear strong convergence), we infer that Λ(, Π Dm u m ) Dm (P Dm ϕ) Λ(, u) ϕ strongly in L 2 () d. By symmetry of Λ and the weak-strong convergence property (Lemma C.3), this shows that Λ(x, Π Dm u m (x)) Dm u(x) Dm (P Dm ϕ)(x)dx = Dm u(x) [Λ(x, Π Dm u m (x)) Dm (P Dm ϕ)(x)] dx u(x) [Λ(x, u(x)) ϕ(x)] dx as m

78 = 3.2 Unknown-dependent diffusion problems 67 Λ(x, u(x)) u(x) ϕ(x)dx. (3.49) Moreover, since Π Dm (P Dm ϕ) ϕ in L 2 () as m, f(x)π Dm (P Dm ϕ)(x)dx f(x)ϕ(x)dx as m. (3.5) Letting v = P Dm ϕ in (3.44), we can use (3.49) and (3.5) to pass to the limit and see that u is a solution to (3.43). Step 4: strong convergence of Dm u m. Let now prove that Dm u m converges to u in L 2 () d. We let v = u m in (3.44) and we pass to the limit in the right-hand side. Since u is a solution to (3.43), we obtain lim m Λ(x,Π Dm u m (x)) Dm u m (x) Dm u m (x)dx = f(x)u(x)dx = Λ(x, u(x)) u(x) u(x)dx. (3.51) We have Λ(x,Π Dm u m (x))( Dm u m (x) u(x)) ( Dm u m (x) u(x))dx = Λ(x, Π Dm u m (x)) Dm u m (x) Dm u m (x)dx Λ(x, Π Dm u m (x)) Dm u m (x) u(x)dx Λ(x, Π Dm u m (x)) u(x) ( Dm u m (x) u(x))dx. (3.52) By (3.51), the weak convergence of Dm u m, the strong convergence of Λ(, Π Dm u m ) u (obtained by non-linear strong convergence property, see Lemma C.4 page 46), and the weak-strong convergence lemma (Lemma C.3), we infer that Λ(x, Π Dm u m (x))( Dm u m (x) u(x)) ( Dm u m (x) u(x))dx as m. The coercivity of Λ shows that the left-hand side is larger than λ Dm u m (x) u(x)) 2 dx. This quantity therefore converges to and the proof of the strong L 2 () convergence of the gradients is complete.

79 68 3 Elliptic problems Non-homogeneous Dirichlet boundary conditions We again refer to Section for the properties of the trace operator in the case of domain with Lipschitz boundary, and we consider Problem (3.41a), replacing the homogeneous Dirichlet boundary condition by under Assumptions (3.42) and u = g on, g H 1/2 ( ). (3.53) Under these hypotheses, a weak solution of this problem is a function u (again not necessarily unique) satisfying: u {w H 1 (), γ(w) = g}, v H 1 (), Λ(x, u(x)) u(x) v(x)dx = f(x)v(x)dx, and it is approximated by the following gradient scheme. (3.54) Definition 3.17 (Non-linear problem, Dirichlet BCs). If D = (X D = X D, X D,, I D,, Π D, D ) is a GD in the sense of Definition 2.18, then we define the related gradient scheme for (3.54) by Find u I D, g + X D, such that, for any v X D,, Λ(x, Π D u(x)) D u(x) D v(x)dx = f(x)π D v(x)dx. (3.55) This scheme is again leading to a nonlinear system of equations under the form A(u)U = B, similar to (3.45). We then have the following convergence result. Theorem 3.18 (Convergence). Under assumptions (3.42)-(3.53), let ( ) m N be a sequence of GDs in the sense of Definition 2.18, which is space-consistent, limit-conforming and compact in the sense of Definitions 2.2, 2.6 and 2.8 (it is then coercive in the sense of Definition 2.2, see Lemma 2.9). Then, for any m N, there exists at least one u m X Dm solution to the gradient scheme (3.55) and, up to a subsequence, Π Dm u m converges strongly in L 2 () to a solution u of (3.54) and Dm u m converges strongly in L 2 () d to u as m. In the case where the solution u of (3.54) is unique, then the whole sequence converges to u as m in the senses above. Proof. Let us first consider any GD D in the sense of Definition We consider any lifting ḡ H 1 () such that γḡ = g, and we define

80 P Dḡ = 3.2 Unknown-dependent diffusion problems 69 argmin v I D, g+x D, ( ΠD v ḡ L 2 () + D v ḡ L2 () d ). Thanks to Definition 2.2, we get that Π Dm P ḡ converges strongly in L 2 () to ḡ and Dm P ḡ converges strongly in L 2 () d to ḡ. Then, for any solution u to (3.55), writing w = u P Dḡ X D,, we have v X D,, Λ(x, Π D (w + PDḡ)(x)) D w(x) D v(x)dx = f(x)π D v(x)dx Λ(x, Π D (w + PDḡ)(x)) D PDḡ(x) D v(x)dx. The remaining of the proof is then similar to that of Theorem 3.16, reasoning on w instead of u Homogeneous Neumann boundary conditions We consider Problem (3.41a), replacing the homogeneous Dirichlet boundary condition by Λ(, u) u n = on, where n is the unit normal outward to, assumed to be Lipschitz, under Assumptions (3.42) and f(x)dx =. (3.56) Under these hypotheses, again defining H 1 () = {ϕ H 1 (), ϕ(x)dx = }, a weak solution of this problem is a function u (not necessarily unique) satisfying: u H (), 1 v H (), 1 Λ(x, u(x)) u(x) v(x)dx = f(x)v(x)dx, and it is approximated by the following gradient scheme. (3.57) Definition 3.19 (Non-linear problem, homogeneous Neumann case). If D = (X D, Π D, D ) is a GD in the sense of Definition 2.24, then we define the related gradient scheme for (3.57) by Find u X D such that for any v X D, Λ(x, Π D u(x)) D u(x) D v(x)dx + Π D u(x)dx Π D v(x)dx = f(x)π D v(x)dx. (3.58)

81 7 3 Elliptic problems This scheme is again leading to a nonlinear system of equations under the form A(u)U = B, similar to (3.45). But the matrix such obtained is in general full, and equivalent algebraic methods leading to sparse matrices must be used. We then have the following convergence result. Theorem 3.2 (Convergence). Assume (3.42)-(3.56), and let ( ) m N be a sequence of GDs in the sense of Definition 2.24, which is space-consistent, limit-conforming and compact in the sense of Definitions 2.27, 2.28 and 2.29 (it is then coercive in the sense of Definition 2.26). Then, for any m N, there exists at least one u m X Dm solution to the gradient scheme (3.58) and, up to a subsequence, Π Dm u m converges strongly in L 2 () to a solution u of (3.57) and Dm u m converges strongly in L 2 () d to u as m. In the case where the solution u of (3.57) is unique, then the whole sequence converges to u as m in the senses above. Proof. We proceed as in the proof of Theorem For any GD D in the sense of Definition 2.32, let w X D be given, and let u X D be such that Λ(x, Π D w(x)) D u(x) D v(x)dx (3.59) + Π D u(x)dx Π D v(x)dx = f(x)π D v(x)dx, v X D. Then, letting v = u in (3.59), and applying the Cauchy-Schwarz inequality and the coercivity property in the sense of Definition 2.26, we get This shows that min(λ, 1) u 2 D f L 2 () Π D u L 2 () C D f L 2 () u D. u D C D min(λ, 1) f L 2 (). (3.6) Therefore, u is obtained by the resolution of an invertible square linear system (since a null right hand side implies u = ). The mapping w u is continuous, by continuity of the coefficients of the inverse of a matrix with respect to its coefficients. Applying the Brouwer theorem (Theorem C.2), we see that this mapping w u has at least one fixed point. This shows the existence of at least one discrete solution to (3.58). It is clear that any solution to this scheme satisfies (3.6). We denote by u m X Dm such a solution for D =. The estimate (3.6) shows that ( u m Dm ) m N is bounded and thus, up to a subsequence still denoted by (u m ) m N, we find u H 1 () such that Π Dm u m converges strongly in L 2 () and a.e. to u and Dm u m converges weakly in L 2 () d to u. We used here Remark 2.42 and the compactness of the sequence of GDs. We define P D : H 1 () X D by

82 3.2 Unknown-dependent diffusion problems 71 P D ϕ = argmin ( ) ΠD v ϕ L 2 () + D v ϕ L2 (). d (3.61) v X D By space-consistency of the sequence of GDs, for any ϕ H 1 () we have Π Dm (P Dm ϕ) ϕ strongly in L 2 () and Dm (P Dm ϕ) ϕ strongly in L 2 () d. Since 1 (the characteristic function of ) belongs to H 1 (), we can take v = P Dm 1 in (3.58) and pass to the limit. We get, thanks to Hypothesis (3.56), that ( = lim m ( = u(x)dx ) ( ) Π Dm u m (x)dx Π Dm (P Dm ϕ)(x)dx ). This shows that u H () 1 and that Π Dm u m (x)dx =. (3.62) lim m Let ϕ H () 1 be given. Using the non-linear strong convergence property of Lemma C.4 page 46, Λ(, Π Dm u m ) Dm (P Dm ϕ) Λ(, u) ϕ strongly in L 2 () d. Lemma C.3 (weak-strong convergence property) enables us to pass to the limit in (3.58) with v = P Dm ϕ, which proves that u is a solution to (3.57). By passing to the limit in the left-hand side of (3.58) with v = u m and using (3.62), we get lim Λ(x, Π Dm u m (x)) Dm u m (x) Dm u m (x)dx m = f(x)u(x)dx = Λ(x, u(x)) u(x) u(x)dx and the strong convergence of Dm u m to u follows from this as in the proof of Theorem Non-homogeneous Neumann boundary conditions We again refer to Section for the properties of the trace operator in the case of domain with Lipschitz boundary. We consider Problem (3.41a), replacing the homogeneous Dirichlet boundary condition by Λ(, u) u n = h on, where n is the unit normal outward to, assumed to be Lipschitz, under Assumptions (3.42) and

83 72 3 Elliptic problems h L 2 ( ), Under these hypotheses, again defining H () 1 = {ϕ H 1 (), f(x)dx + h(x)ds(x) =. (3.63) ϕ(x)dx = }, a weak solution of this problem is a function u (not necessarily unique) satisfying: u H (), 1 v H (), 1 Λ(x, u(x)) u(x) v(x)dx = f(x)v(x)dx + h(x)γ(v)(x)ds(x). (3.64) We again recall that, owing to Hypothesis (3.63), Problem (3.64) is equivalent to u H 1 (), v H 1 (), Λ(x, u(x)) u(x) v(x)dx + u(x)dx v(x)dx (3.65) = f(x)v(x)dx + h(x)γ(v)(x)ds(x), since letting v 1 in (3.65) implies that u(x)dx =. This problem is therefore approximated by the following gradient scheme. Definition 3.21 (Non-linear problem, Neumann case). If D = (X D, Π D, T D, D ) is a GD in the sense of Definition 2.32, then we define the related gradient scheme for (3.64) by Find u X D such that, for any v X D, Λ(x, Π D u(x)) D u(x) D v(x)dx + Π D u(x)dx = f(x)π D v(x)dx + h(x)t D (v)(x)ds(x). We then have the following convergence result. Π D v(x)dx (3.66) Theorem 3.22 (Convergence). Under assumptions (3.42)-(3.63), let ( ) m N be a sequence of GDs in the sense of Definition 2.32, which is space-consistent, limit-conforming and compact in the sense of Definitions 2.27, 2.34 and 2.36 (it is then coercive in the sense of Definition 2.33). Then, for any m N, there exists at least one u m X Dm solution to the gradient scheme (3.66) and, up to a subsequence, Π Dm u m converges strongly in L 2 () to a solution u of (3.64) and Dm u m converges strongly in L 2 () d to u as m. In the case where the solution u of (3.64) is unique, then the whole sequence converges to u as m in the senses above.

84 3.2 Unknown-dependent diffusion problems 73 Proof. We follow the same line as that of the proof of Theorem 3.2, in addition to the use of Remark 2.41 for the weak convergence of the discrete trace. We first get, thanks again to Brouwer s fix-point theorem, the existence of at least one discrete solution u m X Dm to (3.66). Thanks to the coercivity hypothesis (Definition 2.33) which involves the discrete trace, we then get that C P u m Dm min(λ, 1) ( f L 2 () + h L 2 ( )). (3.67) Then the same arguments as those used in the proof of Theorem 3.2 show that there exists u H 1 (), and a subsequence such that Dm u m converges weakly in L 2 () d to u and Π Dm u m converges weakly in L 2 () to u. Moreover, for the interpolation v m = P Dm ϕ defined by (3.61), for any ϕ H 1 (), we get using Remark 2.41, that T Dm v m γϕ weakly in L 2 ( ). The remaining of the proof is then similar to that of the proof of Theorem Non-homogeneous Fourier boundary conditions We consider the same hypotheses on as in the preceding section (in particular, it is assumed to have a Lipschitz boundary), and we then consider Problem (3.41a) with Fourier boundary conditions: under Assumptions (3.42) and Λ(, u) u n + bu = h on, h L 2 ( ), b L ( ) and there exists b > such that b(x) b for a.e. x. A weak solution of this problem is: u H 1 (), v H 1 (), Λ(x, u(x)) u(x) v(x)dx + b(x)γ(u)(x)γ(v)(x)ds(x) = f(x)v(x)dx + h(x)γ(v)(x)ds(x). This problem is then approximated by the following gradient scheme. (3.68) (3.69) Definition 3.23 (GS for the non-linear problem, non-homogeneous Fourier case). If D = (X D, Π D, T D, D ) is a GD in the sense of Definition 2.48, then we define the related gradient scheme for (3.69) by Find u X D such that, for any v X D, Λ(x, Π D u(x)) D u(x) D v(x)dx + b(x)t D u(x)t D v(x)ds(x) = f(x)π D v(x)dx + h(x)t D (v)(x)ds(x). (3.7)

85 74 3 Elliptic problems The convergence result is similar to the previous ones. Theorem 3.24 (Convergence, Fourier BCs). Under assumptions (3.42)- (3.69), let ( ) m N be a sequence of GDs in the sense of Definition 2.48, which is space-consistent, limit-conforming and compact in the sense of Definitions 2.49, 2.34 and 2.36 (it is then coercive in the sense of Definition 2.33). Then, for any m N, there exists at least one u m X Dm solution to the gradient scheme (3.7) and, up to a subsequence, Π Dm u m converges strongly in L 2 () to a solution u of (3.69), Dm u m converges strongly in L 2 () d to u and T Dm u m converges strongly in L 2 ( ) to γu as m. In the case where the solution u of (3.69) is unique, then the whole sequence converges to u as m in the senses above. Proof. The proof is very similar to the proof of Theorems 3.16 and 3.2, we only indicate here the elements which differ. Letting u = v in (3.7), by assumption on Λ and b and Definition 2.48 of D we obtain min(λ, b) u 2 D Λ(x, Π D u(x)) D u(x) D u(x)dx + b(x)t D u(x)t D u(x)ds(x) = f(x)π D u(x)dx + h(x)t D (u)(x)ds(x) f L2 () Π Du L2 () + h L 2 ( ) T D(u) L2 ( ) C D ( f L 2 () + h L 2 ( ) ) u D. This gives an estimate on u D which allows us, as in the proof of Theorem 3.16, to use Brouwer s fixed point theorem to prove the existence of a solution to (3.7). This estimate also shows that the solution u m for D = is such that u m Dm remains bounded and therefore, using Lemma 2.4 and the compactness of the GDs, that, for some u H 1 (), Π Dm u m u strongly in L 2 () and a.e., T Dm u m γu weakly in L 2 ( ) and Dm u m u weakly in L 2 () d. Defining then P D : H 1 () X D by (3.71) P D ϕ = argmin v X D ( ΠD w ϕ L 2 () + D v ϕ L 2 () d + T D v γϕ L2 () d ) the space-consistency of the sequence of GDs shows that, for any ϕ H 1 (), Π Dm (P Dm ϕ) ϕ strongly in L 2 (), Dm (P Dm ϕ) ϕ strongly in L 2 () d and T Dm (P Dm ϕ) γϕ strongly in L 2 ( ).

86 3.3 p-laplacian type problems: p (1, + ) 75 We can then, as in the proof of Theorem 3.16, use v = P Dm ϕ in (3.7) and pass to the limit, thanks to these strong convergences and to (3.71), to see that u is a solution to (3.69). We then take v = u m in (3.7) and pass to the limit to obtain ( lim m = = Λ(x, Π Dm u m (x)) Dm u m (x) Dm u m (x)dx ) + b(x)t Dm u m (x) 2 ds(x) f(x)u(x)dx + h(x)γu(x)ds(x) Λ(x, u(x)) u(x) u(x)dx + b(x)γu(x) 2 ds(x). This limit and (3.71) allows us to see that Λ(x, Π Dm u m (x))( Dm u m (x) u(x)) ( Dm u m (x) u(x))dx + b(x)(t Dm u m (x) γu(x)) 2 ds(x). By Assumptions (3.42b) on Λ and (3.68) on b, the left-hand side of this limit is greater than min(λ, b)( Dm u m u 2 L 2 () + T d u m γu 2 L 2 ( ) d). The strong convergences of the reconstructed gradient and trace therefore follow. Remark In the linear case (Λ independent of u), it is very easy to obtain error estimates for (3.7) similar to the ones in Theorem 3.2 but with an additional error estimate on the traces. 3.3 p-laplacian type problems: p (1, + ) After the study of the GDM approximation of quasilinear elliptic problem in Section 3.2, we now turn to PDEs involving a nonlinearity with respect to the gradient of the unknown function. We first consider the p-laplace problem, for which error estimate in terms of W D and S D are established, in the same way as error estimates are provided in Section 3.1. Then, in Section 3.3.2, the case of a general Leray Lions operator is considered, and the convergence of the GDM is proved for this model. As mentioned in the introduction of this monograph, some of the techniques developed for the GDM are applicable outside this framework. As an illustration of this, we note that the techniques used here to analyse the GDM for p-laplacian type problems have been adapted in [28, 29] to a high-order numerical scheme not presented under the form of a gradient scheme. These two articles also develop discrete functional analysis results for this high-order scheme, similar to the ones developed in Chapter B in the appendix.

87 76 3 Elliptic problems An error estimate for the p-laplace problem We consider in this section a particular case of a non-linear Leray-Lions problem, the so-called p-laplace equation: div( u p 2 u) = f + div(f ) in, with boundary conditions u = on, under the following assumptions: is an open bounded connected subset of R d (d N ), p (1, + ) f L p (), F L p () d. (3.72a) (3.72b) (3.73a) (3.73b) (3.73c) Under these hypotheses, the weak solution of (3.72) is the unique function u satisfying: u W 1,p () and, for all v W 1,p (), u p 2 u(x) v(x)dx = f(x)v(x)dx F (x) v(x)dx. (3.74) Definition 3.26 (Gradient scheme for the p-laplace problem). Let D = (X D,, Π D, D ) be a GD in the sense of Definition 2.1. The corresponding gradient scheme for Problem (3.74) is defined by Find u X D, such that, for any v X D,, D u(x) p 2 D u(x) D v(x)dx = f(x)π D v(x)dx F (x) D v(x)dx. (3.75) The following lemma establishes the existence and uniqueness of the solutions to (3.74) and (3.75), as well as estimates on these solutions. Lemma Under Hypotheses (3.73), there exists one and only one solution to each of the problems (3.74) and (3.75). These solutions moreover satisfy u Lp () (C P,p f d L p () + F L p () d) 1 p 1 (3.76) and D u D Lp () d (C D f L p () + F L p () d) 1 p 1, (3.77) where C P,p is the continuous Poincaré s constant in W 1,p (), and C D is defined by (2.1).

88 3.3 p-laplacian type problems: p (1, + ) 77 Proof. The existence and uniqueness of u and u D are obtained by noticing that (3.74) and (3.75) are respectively equivalent to the minimisation problems ( ) 1 u argmin v p dx f(x)v(x)dx + F (x) v(x)dx v W 1,p () p (3.78) and ( 1 u D argmin v X D, p D v(x) p dx This equivalence is a consequence of the inequality ) f(x)π D v(x)dx + F (x) D v(x)dx. (3.79) χ, ξ R d, χ + ξ p χ p p χ p 2 χ ξ, which follows by writing that the convex mapping H : ζ ζ p lies above its tangent at χ, and by noting that H(χ) = p χ p 2 χ. The existence and uniqueness of the solutions to (3.78) and (3.79) are classical consequence of standard convex minimisation theorems, see e.g. [6]. Then inequalities (3.76) and (3.77) follow by taking, in each corresponding problem, the solution itself as a test function. Theorem 3.28 (Control of the approximation error). Under Hypotheses (3.73), let u W 1,p () be the solution of Problem (3.74), let D be a GD in the sense of Definition 2.1, and let u D X D, be the solution to the gradient scheme (3.75). Then there exists C 2 >, depending only on p such that: 1. If p (1, 2], u D u D Lp () S [ D(u) + C d 2 WD ( u p 2 u + F ) + S D (u) p 1] [S D (u) p + [ (C D + C P,p ) f L p () + F ] p ] 2 p 2 L p () d p 1. (3.8) 2. If p (2, + ), u D u D Lp () d S D(u) + C 2 [W D ( u p 2 u + F ) + S D (u) [ (C P,p f L p () + F L p () d) 1 p 1 + SD (u) ] p 2 ] 1 p 1. (3.81) As a consequence of (3.8) (3.81), we have the following error estimate: u Π D u D L p () S D (u) + C D (S D (u) + u D u D Lp () d). (3.82)

89 78 3 Elliptic problems Remark 3.29 (Mesh-based gradient schemes). As in Remark 3.3 (for the case p = 2 and d 3), under non-degeneracy assumptions on the meshes, for many mesh-based gradient schemes it can be proved that there exists C R +, not depending on D, such that and ϕ W 2,p () W 1,p (), S D (ϕ) Ch D ϕ W 2,p () ϕ W 1,p () d, W D (ϕ) Ch D ϕ W 1,p () d, where h easures the mesh size. Under the condition p > d/2, the proofs of these inequalities are done, for some important examples of gradient schemes, in Part III. Therefore, in the case where u W 2,p () and u p 2 u + F W 1,p () d, the preceding theorem gives an error estimate of the form O(h p 1 D ) if p (1, 2] and O(h 1/(p 1) D ) if p 2. Proof. We notice that, since f L p (), the equation (3.72a) in the sense of distributions (i.e. taking v Cc () in (3.74)) shows that ϕ = u p 2 u+f belongs to W div,p () defined by (2.5), with divϕ = f. We can therefore take ϕ in the definition (2.6) of W D and we obtain, for any v X D,, denoting by W = W D ( u p 2 u + F ) and S = S D (u), D v(x) ( u(x) p 2 u(x) + F (x)) Π D v(x)f(x)dx D v Lp () d W. We use the fact that u D satisfies (3.75) to replace the term Π D v f and we get We set D v(x) [ u(x) p 2 u(x) D u(x) p 2 D u(x) ] dx and we obtain D v Lp () d W. P D u = argmin w X D, ( Π D w u L 2 () + D w u L2 () d), A(v):= D v(x) [ D P D u(x) p 2 D P D u(x) D u(x) p 2 D u D (x) ] dx D v L p () W d + D v(x) [ D P D u(x) p 2 D P D u(x) u(x) p 2 u(x) ] dx

90 3.3 p-laplacian type problems: p (1, + ) 79 ] D v L p () [W + d D P D u p 2 D P D u u p 2 u L p (). Case p (1, 2]. Thanks to (3.86) in Lemma 3.3 below, we get the existence of C 3 depending only on p such that D P D u p 2 D P D u u p 2 u p L p () d C 3 D P D u u p L p () d, which leads, recalling the definition (2.2) of S D, to A(v) D v L p () d [ W + C p 1 3 S p 1]. (3.83) We then apply (3.88) in Lemma 3.3 with ξ = D P D u and χ = D u D, and use Hölder s inequality with exponents 2/p and 2/(2 p). Taking v = P D u u D, we get C 4 depending only on p such that D P D u D u D p L p () d and thus C 4 A(P D u u D ) p 2 ( D P D u p L p () d + D u D p L p () d ) 2 p 2, D P D u D u D 2 L p () d C 2 p 4 A(P Du u D )( D P D u p L p () d + D u D p L p () d ) 2 p p. Plugging (3.83) into this estimate gives C 5 depending only on p such that D P D u D u D Lp () C [ p 1] d 5 W + S [ D P D u p L p () d + D u D p L p () d ] 2 p 2. We have u D u D L p () S + DP d D u D u D Lp () d, and Estimate (3.8) therefore follows from (3.76) and (3.77). Case p (2, + ). We use (3.85) in Lemma 3.3, Hölder s inequality with exponents p/p = p 1 and p 1 p 2, and (a + b)θ 2 θ (a θ + b θ ) with θ = p p 2, a = DP D u p 2 and b = u p 2. This gives C 6 depending only on p such that D P D u p 2 D P D u u p 2 u L p () d This leads to C 6 D P D u u L p () d( DP D u L p () d + u L p () d)p 2.

91 8 3 Elliptic problems A(v) D v L p () d [ W + C 6 S [ D P D u L p () d + u L p () d ] p 2 ]. (3.84) As before, we take v = P D u u D. Thanks to (3.89) in Lemma 3.3, we get the existence of C 7 depending only on p such that Using (3.84) we infer D P D u D u D p L p () d C 7 A(P D u u D ). D P D u D u D p 1 L p () d C 7 [W + C 6 S [ u L p () d + S] p 2 ], and the proof of (3.81) is complete by invoking (3.76). In the following lemma, we gather a few useful estimates. Lemma 3.3. Let p (1, + ) and d N. Then ξ, χ R d, ξ p 2 ξ χ p 2 χ max(1, p 1) ξ χ ( ξ p 2 + χ p 2 ), (3.85) which implies p (1, 2], ξ, χ R d, ξ p 2 ξ χ p 2 χ 5 ξ χ p 1. (3.86) Moreover, setting C (p) = 2 p 1 for p (1, 2] and C (p) = 2 p 1 for p > 2, there holds ξ, χ R d, which implies and C (p)( ξ p 2 ξ χ p 2 χ) (ξ χ) ξ χ 2 ( ξ + χ ) p 2, (3.87) p (1, 2], ξ, χ R d, ( ) p 2 ξ χ p p 1 ( ξ p 2 ξ χ p 2 2 χ) (ξ χ) (3.88) (2 p 1 ( ξ p + χ p )) 2 p 2, p 2, ξ, χ R d, ξ χ p 2 p 1 ( ξ p 2 ξ χ p 2 χ) (ξ χ). (3.89) Proof. Estimates (3.85) and (3.86) originally appeared in [7]. Let H(ξ) = ξ p 2 ξ. If p 2 then H C 1 (R d ) d and DH(ξ) (p 1) ξ p 2 (where DH is the differential of H and DH the norm induced by the Euclidean norm). Hence, for all ξ, χ R d,

92 3.3 p-laplacian type problems: p (1, + ) 81 H(ξ) H(χ) ξ χ (p 1) max ζ [ξ,χ] ζ p 2. (3.9) The proof of (3.85) is complete in the case p 2 since the mapping s s p 2 is non-decreasing, and thus max ζ [ξ,χ] ζ p 2 = max( ξ p 2, χ p 2 ) ( ξ p 2 + χ p 2 ). If p < 2, (3.9) remains valid but does directly lead to (3.85). Without loss of generality, we can assume that < χ ξ (the case where χ = is trivial since the right-hand side of (3.85) is then equal to + ). Let ξ be the point in R d at the intersection of the segment (ξ, χ) and of the ball of centre and radius χ (see Figure 3.1). Since ξ = χ we have H( ξ ) H(χ) = χ p 2 ξ χ. Hence, by the triangular inequality and (3.9) between ξ and ξ, H(ξ) H(χ) H(ξ) H( ξ) + χ p 2 ξ χ ξ ξ (p 1) max ζ p 2 + χ p 2 ξ χ. ζ [ξ, ξ] Since p < 2, max ζ [ξ, ξ] ζ p 2 = ξ p 2 = χ p 2 and therefore H(ξ) H(χ) [ ξ ξ ] + ξ χ χ p 2. The proof of (3.85) in the case p < 2 is complete by noticing that ξ ξ + ξ χ = ξ χ. ξ χ 11 ξ Fig Illustration of the proof of (3.85) in the case p (1, 2). Let us now prove (3.86). Let η >. If ξ and χ belong to [η, + ), by (3.85) we have ξ p 2 ξ χ p 2 χ 2η p 2 ξ χ. (3.91) Otherwise, assume that χ (, η]. We have ξ p 2 ξ χ p 2 χ ξ p 1 + η p 1 ( ξ χ + η) p 1 + η p 1. (3.92) Combining (3.91) and (3.92) we see that, for all ξ, χ R d and all η >,

93 82 3 Elliptic problems ξ p 2 ξ χ p 2 χ 2η p 2 ξ χ + ( ξ χ + η) p 1 + η p 1. Estimate (3.86) follows by choosing η = ξ χ. We now turn to the proof of (3.87). Set A = ( ξ p 2 ξ χ p 2 χ) (ξ χ). By developing both sides we see that A = ( ξ p 1 χ p 1 )( ξ χ ) + ( ξ p 2 + χ p 2 )( ξ χ ξ χ). (3.93) Let us prove that the function f(x) = x p 1 y p 1 2 C (p) (x + y)p 2 (x y) satisfies f (x) for all x y. We have f (x) = (p 1)x p 2 2 C (p) (p 2)(x + y)p 3 (x y) 2 C (p) (x + y)p 2. If 1 < p 2, we write f (x) (p 1)x p 2 2 C (p) xp 2 = since C (p) = 2 p 1. If p > 2, f (x) (p 1)x p 2 2 C (p) (p 2)(x+y)p 3 (x+y) 2 C (p) (x+y)p 2, and therefore, since x y, f (x) (p 1)(x p 2 2 C (p) 2p 2 x p 2 ) = since C (p) = 2 p 1. Since f(y) =, this shows that, if x y, x p 1 y p 1 2 C (p) (x + y)p 2 (x y). Assuming (without loss of generality) that ξ χ and applying the previous inequality to x = ξ and y = χ gives ( ξ p 1 χ p 1 )( ξ χ ) 1 C (p) ( ξ + χ )p 2 ( ξ χ ) 2. (3.94) Let us again take generic numbers x y. If 1 < p 2 we can write x p 2 + y p 2 y p 2 (x + y) p 2 (p 1)(x + y) p 2 = 2 C (p) (x + y)p 2. If p > 2 we have x p 2 + y p 2 x p p (x + y) p 2 = 2 C (p) (x + y)p 2. Applying these inequalities with x = ξ and y = χ, plugging the result in (3.93) and using (3.94) leads to A 1 C (p) ( ξ + χ )p 2 [ ( ξ χ ) 2 + 2( ξ χ ξ χ) ]. The proof of (3.87) is complete by writing

94 3.3 p-laplacian type problems: p (1, + ) 83 ( ξ χ ) 2 + 2( ξ χ ξ χ) = ξ 2 2 ξ χ + χ ξ χ 2ξ χ = ξ 2 2ξ χ + χ 2 = ξ χ 2. Estimate (3.88) is obtained by raising (3.87) to the power p/2 and by using ( ξ + χ ) p 2 p 1 ( ξ p + χ p ). Estimate (3.89) follows by writing ξ χ p = ξ χ 2 ξ χ p 2 ξ χ 2 ( ξ + χ ) p 2 and by using (3.87) Convergence of gradient schemes for fully nonlinear Leray Lions problems We now study the convergence of gradient schemes for the non-linear problem div a(x, u, u) = f in, u = on, (3.95) under the following assumptions: p (1, ) and a : L p () R d R d is a Caratheodory function (3.96a) (i.e. for a.e. x the function (u, ξ) a(x, u, ξ) is continuous, and for any (u, ξ) L p () R d the function x a(x, u, ξ) is measurable), a (, + ) such that a(x, u, ξ) ξ a ξ p for a.e. x, u L p (), ξ R d, (a(x, u, ξ) a(x, u, χ)) (ξ χ) for a.e. x, u L p (), ξ, χ R d, a L p (), µ (, + ) such that a(x, u, ξ) a(x) + µ ξ p 1 for a.e. x, u L p (), ξ R d, f L p (), where p = p p 1. (3.96b) (3.96c) (3.96d) (3.96e) Remark Note that the dependence of a on u is assumed to be non-local: a(x, u, ) depends on all the values of u L p (), not only on u(x). These assumptions cover for example the case where a(x, u, u(x)) = Λ[u](x) u(x) with Λ : L p () L (; S d (R)) as in [18, 3, 71]. These assumptions (in particular (3.96a)) do not cover the usual local dependencies a(x, u(x), u(x)) as in the non-monotone operators studied in [64]. However, the adaptation of the following results to this case is quite easy and more classical, see e.g. [32] for an adaptation of the original Leray-Lions method to a numerical scheme (based on the Mixed Finite Volume method) for local non-monotone operators.

95 84 3 Elliptic problems If a function a satisfies (3.96), then the mapping u diva(, u, u) is called a generalised Leray-Lions operator. A classical example is the p-laplacian operator, obtained by setting a(x, u, ξ) = ξ p 2 ξ. Note that the existence of at least one solution to (3.95) is shown in [64] under hypotheses (3.96) in the case where a does not depend on u. In our framework, we say that a function u is a weak solution to (3.95) if: u W 1,p (), v W 1,p (), a(x, u, u(x)) v(x)dx = f(x)v(x)dx. (3.97) Remark Note that, even if a does not depend on u L p (), the solution to (3.97) is not necessarily unique. Consider the case where d = 1, = ( 1, 2), f(x) = for x ( 1, ) (1, 2), f(x) = 2 for x (, 1) and a(x, u, ξ) = (min( ξ, 1) + max( ξ 2, )) ξ ξ, ξ R, u L2 (). Then (3.96b) is satisfied with a = 1 2, (3.96c) is satisfied since a is non decreasing with respect to ξ and (3.96d) is satisfied with a(x) = and µ = 1. Then the function u(x) = α(x + 1) for x ( 1, ), α + x(1 x) for x (, 1), α(2 x) for x (1, 2) is solution to (3.97) for any value α [1, 2]. The hypothesis that a is strictly monotone, which may be expressed by (a(x, u, ξ) a(x, u, χ)) (ξ χ) >, for a.e. x, u L p (), ξ, χ R d with ξ χ, (3.98) is only used to prove the strong convergence of the approximate gradient (see theorem below). We now define the gradient scheme for Problem (3.95). Definition 3.33 (GS for fully non-linear Leray Lions problems, homogeneous Dirichlet BCs). If D = (X D,, Π D, D ) is a GD, then we define the related gradient scheme for (3.95) by Find u X D, such that, v X D,, a(x, Π D u, D u(x)) D v(x)dx = f(x)π D v(x)dx. (3.99) Theorem 3.34 (Convergence). Under assumptions (3.96)-(3.96e), take a sequence ( ) m N of GDs in the sense of Definition 2.1, which is spaceconsistent, limit-conforming and compact in the sense of Definitions 2.4, 2.6 and 2.8 (it is then coercive in the sense of Definition 2.2). Then, for any m N, there exists at least one u Dm X Dm, solution to the gradient scheme (3.99) and, up to a subsequence, Π Dm u Dm converges strongly in L p () to a solution u of (3.97) and Dm u Dm converges weakly in L p () d to u as m. Moreover, if we assume that the Leray-Lions operator a is

96 3.3 p-laplacian type problems: p (1, + ) 85 strictly monotone in the sense of (3.98), then Dm u Dm converges strongly in L p () d to u as m. In the case where the solution u of (3.97) is unique, then the whole sequence converges to u as m in the above senses. Remark As a by-product, this theorem also gives the existence of a solution ū to (3.97). Indeed, under the assumptions of the theorem, the proof shows that the sequence u Dm has a converging subsequence and that the limit ū of this subsequence is in fact a solution to the continuous problem. Since there exists at least one (in fact there exist several, see Part III) gradient scheme which satisfies the assumptions of this theorem, this gives the existence of a solution to (3.97). Remark 3.36 (Non-linearity without a lower order term) In the case where a does not depend on u L p (), the proof of the weak convergence of Π Dm u to a solution of (3.97) does not require the compactness of the sequence of GDs. In this case the strong convergence results from (3.98) (which gives the strong convergence of the approximate gradient) and from the coercivity and the space-consistency of the sequence () m N. Proof. This proof follows the same ideas as in [32, 5]. Step 1: existence of a solution to the scheme. Let D be a GD in the sense of Definition 2.1. We endow the finite dimensional space X D, with a inner product, and we denote by its related norm. We define F : X D, X D, as the function such that, if u X D,, F (u) is the unique element in X D, which satisfies v X D,, F (u), v = a(x, Π D u, D u(x)) D v(x)dx. Likewise, we denote by w X D, the unique element such that v X D,, w, v = f(x)π D v(x)dx. The assumptions on a show that F is continuous and that, for all u X D,, F (u), u a D u p. By equivalence of the norms and L p () d D Lp () d on X D,, we deduce that F (u), u C 8 u p with C 8 not depending on u. This F (u),u shows that lim u u = + and thus that F is surjective (see [64] or [27, Theorem 3.3, page 19]). Note that we could as well use Theorem C.1, consequence of the topological degree. There exists therefore u D X D, such that F (u D ) = w, and this u D is a solution to (3.99). Step 2: convergence to a solution of the continuous problem.

97 86 3 Elliptic problems Letting v = u Dm in (3.99) with D = and using (2.1) and Hypothesis (3.96b), we get a Dm u Dm p 1 C L p () d Dm f L p (). Thanks to the coercivity of the sequence of GDs, this provides an estimate on Dm u Dm in L p () d and on Π Dm u Dm in L p (). Lemma 2.12 then gives u W 1,p () such that, up to a subsequence, Π Dm u Dm u weakly in L p () and Dm u Dm u weakly in L p () d. By compactness of the sequence of GDs, we can also assume that the convergence of Π Dm u Dm to u is strong in L p () (this strong convergence property is only necessary for coping with the dependence of a with respect to u). By Hypothesis (3.96d), the sequence of functions A Dm (x) = a(x, Π Dm u Dm, Dm u Dm (x)) remains bounded in L p () d and converges therefore, up to a susbsequence, to some A weakly in L p () d, as m. Let us now show that u is solution to (3.97), using the well-known Minty trick [66]. For a given ϕ W 1,p () and for any GD D belonging to the sequence ( ) m N, we introduce ( ) P D ϕ = argmin ΠD v ϕ Lp () + D v ϕ L p () d v X D, as a test function in (3.99). By the space-consistency of ( ) m N, letting m we get A(x) ϕ(x)dx = f(x)ϕ(x)dx, ϕ W 1,p (). (3.1) On the other hand, we may let m in (3.99) with u Dm as a test function. Using (3.1) with ϕ = u, this leads to lim a(x, Π Dm u Dm, Dm u Dm (x)) Dm u Dm (x)dx m (3.11) = f(x)u(x)dx = A(x) u(x)dx. Hypothesis (3.96c) gives, for any G L p () d, (a(x, Π Dm u Dm, Dm u Dm (x)) a(x, Π Dm u Dm, G(x))) ( Dm u Dm (x) G(x))dx. Developing the preceding inequality, using Lemma C.3 for the weak-strong convergences and (3.11) for the convergence of the sole term involving a product of two weak convergences, we may let m and we get

98 3.3 p-laplacian type problems: p (1, + ) 87 (A(x) a(x, u, G(x))) ( u(x) G(x))dx, G L p () d. We then set G = u + αϕ in the preceding inequality, where ϕ Cc () d and α >. Dividing by α, we get (A(x) a(x, u, u(x) + αϕ(x))) ϕ(x)dx, ϕ Cc () d, α >. We then let α and use the dominated convergence theorem, which leads to (A(x) a(x, u, u(x))) ϕ(x)dx, ϕ Cc () d. Changing ϕ into ϕ, we deduce that (A(x) a(x, u, u(x))) ϕ(x)dx =, ϕ Cc () d, and therefore that A(x) = a(x, u, u(x)), for a.e. x. (3.12) In addition to (3.1), this shows that u is a solution to (3.97). This concludes the proof of the convergence of Π Dm u Dm to u in L p () and of Dm u Dm to u weakly in L p () d as m. Step 3: Assuming now Hypothesis (3.98), strong convergence of the approximate gradient. We follow here the ideas of [64]. Thanks to (3.11) and (3.12), we get [ ] lim a(x, Π Dm u Dm (x), Dm u Dm (x)) a(x, Π Dm u Dm (x), u(x)) m [ ] Dm u Dm (x) u(x) dx =. Since the integrand is non-negative, this shows that [a(, Π Dm u Dm, Dm u Dm ) a(, Π Dm u Dm, u)] [ Dm u Dm u] in L 1 (), (3.13) and therefore a.e. for a sub-sequence. Then, thanks to the strict monotonicity assumption (3.98), we may use Lemma 3.37 given below to show that Dm u Dm u a.e. as m, at least for the same sub-sequence. This shows the a.e. convergence of a(, Π Dm u Dm, Dm u Dm ) D u D to a(, u, u) u. We next recall that, by (3.11) and (3.12), lim a(x, Π Dm u Dm, Dm u Dm (x)) Dm u Dm (x)dx m

99 88 3 Elliptic problems = a(x, u, u(x)) u(x)dx. (3.14) Since a(, Π Dm u Dm, Dm u Dm ) Dm u Dm, we can apply Lemma 3.38 to get a(, Π Dm u Dm, Dm u Dm ) Dm u Dm a(, u, u) u in L 1 () as m. This L 1 -convergence gives the equi-integrability of the sequence of functions a(, Π Dm u Dm, Dm u Dm ) Dm u Dm, which gives in turn, thanks to (3.96b), the equi-integrability of ( Dm u Dm p ) m N. The strong convergence of Dm u Dm to u in L p () d is then a consequence of Vitali s theorem. Lemma Let B be a metric space, let b be a continuous function from B R d to R d such that (b(u, δ) b(u, γ)) (δ γ) >, δ γ R d, u B. Let (u m, β m ) n N be a sequence in B R d and (u, β) B R d such that (b(u m, β m ) b(u m, β)) (β m β) and u m u as m. Then, β m β as m. Proof. We begin the proof with a preliminary remark. Let δ R d \{}. We define, for all m N, the function h δ,m from R to R by h δ,m (s) = (b(u m, β + sδ) b(u m, β)) δ. The hypothesis on b gives that h δ,m is an increasing function since, for s > s, one has : h δ,m (s) h δ,m (s ) = (b(u m, β + sδ) b(u m, β + s δ)) δ >. We prove now, by contradiction, that lim m β m = β. If the sequence (β m ) m N does not converge to β, there exists ε > and a subsequence, still denoted by (β m ) m N, such that s m := β m β ε, for all m N. Then, we set δ m = βm β β m β and we can assume, up to a subsequence, that δ m δ as m, for some δ R d with δ = 1. We then have, since s m ε, Then, passing to the limit as m, (b(u m, β m ) b(u m, β)) βm β s m = h δm,m(s m ) h δm,m(ε) = (b(u m, β + εδ m ) b(u m, β)) δ m. 1 = lim (b(u m, β m ) b(u m, β)) (β m β) (b(u, β + εδ) b(u, β)) δ >, m s m which is impossible. The following result is classical (see [64]). Its proof is given for the sake of completeness. Lemma Let (F m ) m N be a sequence non-negative functions in L 1 (). Let F L 1 () be such that F m F a.e. in and F m(x)dx F (x)dx, as m. Then, F m F in L 1 () as m.

100 3.3 p-laplacian type problems: p (1, + ) 89 Proof. The proof of this lemma is very classical. Applying the Dominated Convergence Theorem to the sequence (F F m ) + leads to (F (x) F m (x)) + dx as m. Then, since F F m = 2(F F m ) + (F F m ), we conclude that F m F in L 1 () as m.

101

102 Part II Parabolic problems

103

104 4 Time-dependent problems: gradient discretisation method and discrete functional analysis In this chapter, we first give the definition of gradient discretisations (GDs) for time-dependent problems. We then present compactness results for the analysis of such problems. These results include discrete Ascoli Arzelà and Aubin Simon theorems, and are presented first in an general setting, before their consequences for gradient discretisations are discussed. 4.1 Space time gradient discretisation To fix ideas, let us consider a general time-dependent problem under the form t u + A(u) = f, over a domain (, T ) with T >. Adequate boundary conditions and initial conditions are also assumed. If θ [, 1] and t () = < t (1) < < t (N) = T is a set of time points, then the θ-scheme reads: for all n =,..., N 1, u (n+1) u (n) t (n+1) t (n) + A(θu(n+1) + (1 θ)u (n) ) = f (n). (4.1) For θ = 1, the scheme is Euler implicit (or backward ), for θ = it is Euler explicit (or forward ), and θ = 1 2 provides the Crank-Nicolson scheme. Implicit schemes correspond to θ [ 1 2, 1], and are the most frequently considered in this book due to the parabolic nature of the equations under study. To deal with all kinds of boundary conditions at once, the notation X D, stands for X D, in the case of homogeneous Dirichlet boundary conditions, and for X D in the case of other boundary conditions. Similarly, we write W 1,p () for W 1,p () in the case of homogeneous Dirichlet boundary conditions, and W 1,p () in the case of other boundary conditions. Definition 4.1 (GD for time-dependent problems). Let p (1, + ), be an open subset of R d (with d N ), T > and θ [, 1]. We say that D T = (D, I D, (t (n) ) n=,...,n ) is a space time gradient discretisation if

105 94 4 Time-dependent problems: GDM and DFA D = (X D,, Π D, D,...) is a GD in the sense of Definition 2.1 (resp. Definition 2.18, 2.24, 2.32, 2.48 or 2.51 depending on the considered boundary conditions), which satisfies Π D (X D, ) L max(p,2) (), I D : L 2 () X D, is an interpolation operator, t () = < t (1)... < t (N) = T. The gradient discretisation D is called the underlying spatial discretisation of D T. We then set δt (n+ 1 2 ) = t (n+1) t (n), for n =,..., N 1, and δt D = max n=,...,n 1 δt (n+ 1 2 ). To a family v = (v (n) ) n=,...,n X N+1 D, we associate the functions v θ L (, T ; X D, ), Π (θ) D v L (, T ; L max(p,2) ()), (θ) D v L (, T ; L p () d ) and T (θ) D v L (, T ; L p ( )) defined by n =,..., N 1, for all t (t (n), t (n+1) ], v θ (t) = v (n+θ) := θv (n+1) + (1 θ)v (n) and, for a.e. x, Π (θ) D v(x, t) = Π D[v θ (t)](x), (θ) D v(x, t) = D[v θ (t)](x) and T (θ) D v(x, t) = T D[v θ (t)](x). (4.2) To state uniform-in-time convergence results, we also need to extend the definition of Π (θ) D v up to t = : For a.e. x, Π (θ) D v(x, ) = Π Dv () (x). (4.3) If v X N+1 D,, we define δ Dv L (, T ; L max(p,2) ()) by n =,..., N 1, for a.e. t (t (n), t (n+1) ), δ D v(t) = δ (n+ 1 2 ) D v := Π Dv (n+1) Π D v (n). δt (n+ 1 2 ) (4.4) Remark 4.2. The iterative definition (4.1) requires the initialisation step, a way to compute u (). The interpolation operator I D applied to the initial condition describes this initialisation of u () (cf., e.g., (5.5) in Section 5.1). Definition 4.3 (Space time-consistency for space time GD) For T > and θ [, 1], if D T is a space time GD in the sense of Definition 4.1, we define ŜD by (2.2), (2.14), (2.2), (2.49) or (2.54) (depending on the considered boundary conditions), where W 1,p () has been replaced with W 1,p () L 2 () and Π D v ϕ L p () has been replaced with Π D v ϕ L max(p,2) (). A sequence ((D T ) m ) m N of space time GDs in the sense of Definition 4.1, with underlying spatial discretisations ( ) m N, is said to be space time-consistent if

106 4.1 Space time gradient discretisation It holds ϕ W 1,p () L 2 (), lim Ŝ Dm (ϕ) =. m 2. It holds u L 2 (), 3. δt Dm as m. lim u Π I Dm u m L 2 () =. (4.5) Remark 4.4 (A generic definition of I D) Given a spatial gradient discretisation D such that Π D(X D, ) L 2 (), we can define an interpolator I D : L 2 () X D, by u L 2 (), I Du = argmin { v D : v X D,, Π Dv = Pr ΠD (X D, )u}, (4.6) where Pr ΠD (X D, ) : L 2 () Π D(X D, ) is the L 2 ()-orthogonal projector on Π D(X D, ). Since the norm D is uniformly convex (see the definitions in Chapter 2, depending on the various boundary conditions), the argmin in (4.6) is indeed unique, and we can even check that I D : L 2 () X D, is linear continuous (although this is not required in Definition 4.1). Consider now a sequence () m N of spatial GDs, such that, as m, Ŝ Dm (u) for all u W 1,p () L 2 () (this is an improved consistency property of () m N ). This shows that, for such an u, there exists u m X Dm, such that Π Dm u m u L 2 (). The definition (4.6) yields ΠDm I Dm = Pr ΠDm (X Dm, ), and thus, by the properties of the orthogonal projector, u u Π Dm I Dm u L 2 () = PrΠDm (X Dm, )u L 2 () u Π Dm u m L 2 () as m. Hence, (4.5) holds for u W 1,p () L 2 (). Since the mapping Π Dm I Dm = Pr ΠDm (X Dm, ) : L 2 () L 2 () has norm 1, reasoning by density of W 1,p () L 2 () into L 2 () shows that (4.5) actually holds for all u L 2 (). Remark 4.5. To illustrate the definition of ŜD, here is how it looks for Fourier boundary conditions: ϕ W 1,p () L 2 (), Ŝ D (ϕ) = min v X D ( ΠD v ϕ L max(p,2) () + T Dv γϕ Lp ( ) + D v ϕ Lp () d ).

107 96 4 Time-dependent problems: GDM and DFA The notions of coercivity, limit-conformity and compactness for sequences of space time GDs boil down the corresponding notion for the sequence of underlying spatial discretisations. Definition 4.6 (Coercivity, limit-conformity and compactness for space time GDs) Let T > and ((D T ) m ) m N be a sequence of space time GDs in the sense of Definition 4.1, with underlying spatial discretisations ( ) m N. The sequence ((D T ) m ) m N is coercive (resp. limit-conforming, resp. compact) if the sequence ( ) m N is coercive (resp. limit-conforming, resp. compact) for the corresponding boundary conditions. The following lemma is the counterpart of Lemma 2.12 and Lemma 2.4. We could as easily state counterparts of the regularity of the limit for nonhomogeneous Dirichlet boundary conditions or mixed boundary conditions (as in Lemma 2.23 or Lemma 2.57). Lemma 4.7 (Regularity of the limit, space time problems). Let p (1, ) and ((D T ) m ) m N be a coercive and limit-conforming sequence of space time GDs, in the sense of Definition 4.6, for homogeneous Dirichlet or nonhomogeneous Neumann boundary conditions. Let θ [, 1], q (1, + ) and take, for any m N, u m X Nm+1, such that ( (u m ) θ Lq (,T ;X Dm, ) ) m N is bounded. Then there exists u L q (, T ; W 1,p ()) such that, up to a subsequence, Π (θ) u m u weakly in L q (, T ; L p ()) and (θ) u m u weakly in L q (, T ; L p ()) d. In the case of non-homogeneous Neumann boundary conditions, we also have, up to a subsequence, T (θ) u m γu weakly in L q (, T ; L p ( )). The same conclusions hold in the case q = +, provided that the weak convergences are replaced with weak- convergences. Remark 4.8. Note that each space X Dm, is endowed with its natural norm Dm, i.e. Dm Lp () for Dirichlet boundary conditions and (2.18) for d Neumann boundary conditions. For q < +, a bound on (u m ) θ L q (,T ;X Dm, ) is therefore a bound on ( 1/q T (u m ) θ (t) q dt). Proof. We only consider the case of homogeneous Dirichlet boundary conditions, the other case being handled similarly by following the proof of Lemma 2.4. By coercivity of ((D T ) m ) m N,

108 4.1 Space time gradient discretisation 97 (u m ) θ Lq (,T ;X Dm, ) = (θ) u m Lq (,T ;L p ()) d 1 Π (θ) L D C m u m. P q (,T ;L p ()) The sequences (Π (θ) u m ) m N and ( (θ) u m ) m N are therefore bounded in L q (, T ; L p ()) and L q (, T ; L p ()) d, respectively. Up to a subsequence, they converge weakly (or weakly- if q = + ) in these spaces towards u and v, respectively. Extending all the functions by outside, these convergences still hold weakly in L q (, T ; L p (R d )) and L q (, T ; L p (R d )) d. The proof is completed by showing that v = u in the sense of distributions on R d (, T ). Let ϕ Cc (R d ) d and ψ Cc (, T ). We drop the indices m for legibility. We have, for t (, T ), by definition (2.6) of W D, [ ] D [u θ (t)](x) ϕ(x) + Π D [u θ (t)](x)divϕ(x) dx u θ (t) XD, W D (ϕ). Multiply this by ψ(t), integrate over t (, T ) and use Hölder s inequality: T [ W D (ϕ) (θ) D T u(x, t) (ψ(t)ϕ(x)) + Π(θ) D u(x, t)div(ψ(t)ϕ)(x) ]dxdt u θ (t) XD, ψ(t)dt W D (ϕ) u θ L q (,T ;X D, ) ψ L q (,T ). Changing ψ into ψ shows that the same inequality is satisfied if the lefthand side is replaced with its absolute value. Since ( u θ L q (,T ;X D, ) ) m N is bounded and, by limit-conformity, W D (ϕ) as m, we can pass to the limit and see that T [ ] v(x, t) (ψ(t)ϕ(x)) + u(x, t)div(ψ(t)ϕ)(x) dxdt =. This relation holds true for linear combinations of functions of the form (x, t) ψ(t)ϕ(x), that is for all tensorial smooth functions. These tensorial functions are dense (e.g. for the C 1 ( [, T ]) d norm) in C c (R d (, T )) d, see [31, Appendix D]. This shows that, for all Φ C c (R d (, T )) d, T [ ] v(x, t) Φ(x, t) + u(x, t)div(φ(, t))(x) dxdt =. Hence, v = u in the sense of distributions on R d (, T ), as required. The following result shows that functions depending on time and space can be approximated, along their gradient, with reconstructed functions and gradients built from space time-consistent GDs.

109 98 4 Time-dependent problems: GDM and DFA Lemma 4.9 (Interpolation of space time functions). For p [1, ), T > and θ [, 1], let ((D T ) m ) m N be a sequence of space time GDs in the sense of Definition 4.1, which is space time-consistent in the sense of Definition 4.3. Let v L p (, T ; W 1,p ()). Then: 1. There exists a sequence (v m ) m N such that v m = (v m (n) ) n=,...,nm X Nm+1, for all m N, and, as m, Π (θ) v m v strongly in L p ( (, T )), (θ) v m v strongly in L p ( (, T )) d. (4.7a) (4.7b) 2. In the case of non-homogeneous Neumann boundary conditions, if the sequence of underlying spatial discretisations is coercive and limit conforming in the sense of Definitions 2.33 and 2.34, then the sequence (v m ) m N in Item 1 also satisfies T (θ) v m γv weakly in L p ( (, T )) as m. (4.8) 3. In the case of non-homogeneous Fourier boundary conditions, the sequence (v m ) m N in Item 1 can be chosen such that T (θ) v m γv strongly in L p ( (, T )) as m. (4.9) 4. If moreover v C([, T ]; L 2 ()), t v L 2 ( (, T )) and v(, T ) =, then the sequence (v m ) m N in Item 1 can be chosen such that, in addition to (4.7), m N, v (Nm 1) m = v (Nm) m = (and thus Π (θ) v m = on (t (Nm 1), t (Nm) ]), Π (θ) v m (, ) v(, ) strongly in L 2 () as m, δ Dm v m t v strongly in L 2 ( (, T )) as m. (4.1a) (4.1b) (4.1c) Proof. Step 1: proof of Item 1. Define the set T (, T ; W 1,p ()) of space time tensorial functions the following way: v T (, T ; W 1,p ()) if there exist l N, a family (ϕ i ) i=1,...,l C ([, T ]) and a family (w i ) i=1,...,l W 1,p () such that v(x, t) = l ϕ i (t)w i (x) for a.e. x and all t (, T ). (4.11) i=1 By [31, Corollary 1.3.1], T (, T ; W 1,p ()) is dense in L p (, T ; W 1,p ()) and we can therefore reduce the proof of (4.7) to the case v T (, T ; W 1,p ()) (the proof of this reduction is similar to the proof of Lemma 2.13).

110 4.1 Space time gradient discretisation 99 Given the structure (4.11) of functions in T (, T ; W 1,p ()), we actually only need to prove the result for v(x, t) = ϕ(t)w(x) with ϕ C ([, T ]) and w W 1,p (). Let v m X Nm+1,,..., N m, where be defined by v (n) m = ϕ(t (n) )P Dm w for n = ( ) P Dm w = argmin Π Dm z w L max(p,2) () + z w L p (). d z X Dm, (4.12) Define Φ m : (, T ] R as the piecewise constant function equal to θϕ(t (n+1) )+ (1 θ)ϕ(t (n) ) on (t (n), t (n+1) ] for all n =,..., N m 1. Then, by definition (4.2) of the space time reconstruction operator Π (θ), for all t (, T ) and a.e. x, v(x, t) Π (θ) v m (x, t) = ϕ(t)w(x) Φ m (t)π Dm (P Dm w)(x) (4.13) = [ϕ(t) Φ m (t)]w(x) + Φ m (t)[w(x) Π Dm (P Dm w)(x)]. Using the definitions of Ŝ and P Dm, we infer that v Π (θ) v m L p ( (,T )) ϕ Φ m L p (,T ) w L p () + Φ m L p (,T ) w Π (P Dm w) L p () ϕ Φ m L p (,T ) w L p () + Φ m L p (,T ) Ŝ (w). (4.14) As m, the space time-consistency of ((D T ) m ) m N gives Ŝ (w) and the smoothness of ϕ shows that Φ m ϕ uniformly (and thus in L p (, T )). Hence, (4.7a) follows from (4.14). The proof of (4.7b) is obtained by the same argument starting from (4.13) and replacing v with v, w with w, and Π (θ) with (θ). Step 2: proof of Items 2 and 3. In the case of Neumann boundary conditions, applying Lemma 4.7 to (v m ) m N yields (4.8). In the case of Fourier boundary conditions, the definition (4.12) can be replaced with ( P Dm w = argmin z XDm Π Dm z w L max(p,2) () + z w L p () ) d + T Dm z γw L p ( ) and the reasoning starting from (4.13) can be done with (γv, T (θ) v m ) instead of (v, Π (θ) v m ), since T Dm (P Dm w) γw Lp ( ) Ŝ (w). This shows that (4.9) holds. Step 3: proof of Item 4. Assume that v L p (, T ; W 1,p ()) C([, T ]; L 2 ()), t v L 2 ( (, T )) and v(, T ) =. By [31, Theorem 2.3.1], we can find a sequence (v n ) n N C ([, T ]; W 1,p () L 2 ()) such that, as n,

111 1 4 Time-dependent problems: GDM and DFA v n v in L p (, T ; W 1,p ()) C([, T ]; L 2 ()), and t v n t v in L 2 (, T ; L 2 ()). The proof of [31, Theorem 2.3.1] is based on an even extension of v at t = T, required to preserve the continuity of the extended function. Since v(, T ) =, we can actually use an extension by on [T, ) and, by selecting in [31, Theorem 2.3.1] a smoothing kernel with support in ( T, ), we ensure that each v n vanishes on [T ɛ n, ) for some ɛ n >. Having approximated v by these v n, we just need to prove the result for each v n instead of v. Let us drop the index n and write v for v n. We have v C ([, T ]; W 1,p () L 2 ()) and v = on [T ɛ, T ] for some ɛ >. Let τ (, ɛ/4), l τ = T/τ and take (ψ i ) i=1,...,lτ C ([, T ]) a partition of unity on [, T ] subordinate to the open covering ((iτ 2τ, iτ + 2τ) i=1,...,lτ ). Set l τ ( t ) v τ (x, t) = v(x, T ) + ψ i (s)ds t v(x, iτ) = l τ i=1 i=1 ( t ) ψ i (s)ds t v(x, iτ). T Since v = on [T ɛ, T ] [T 4τ, T ], the terms corresponding to i = l τ 3,..., l τ in the previous sum vanish (since iτ l τ τ 3τ T 4τ). For i l τ 4, the support of ψ is contained in [, T 2τ]. This shows that v τ (, t) = for all t [T 2τ, T ]. We write Since t v τ (x, t) = l τ i=1 l τ i=1 T ψ i (t) t v(x, iτ). we have t v(x, t) = l τ i=1 ψ i(t) t v(x, t) and thus t v τ (, t) t v(, t) W 1,p () L 2 () ψ i (t) = 1 for all t (, T ), (4.15) l τ i=1 ψ i (t) t v(, iτ) t v(, t) W 1,p () L 2 (). Using the fact that ψ i (t) only if t iτ < 2τ (that is, iτ (t 2τ, t+2τ)), and invoking (4.15), we infer t v τ (, t) t v(, t) W 1,p () L 2 () sup r (t 2τ,t+2τ) t v(, r) t v(, t) W 1,p () L 2 ().

112 4.2 Averaged-in-time compactness 11 By smoothness of v, this shows that t v τ t v in L (, T ; W 1,p () L 2 ()) as τ. Integrating and using v τ (, T ) = v(, T ) =, we obtain v τ v in C([, T ]; W 1,p () L 2 ()). Hence, we only need to find approximations in the sense (4.7) and (4.1) for each v τ instead of v. Given the structure of v τ, this amounts to finding such approximations when v(x, t) = ϕ(t)w(x) with w W 1,p () L 2 () and ϕ C ([, T ]) having support in [, T ν] for some ν >. We set, as before, v m (n) = ϕ(t (n) )P Dm w for n =,..., N m. The proof of (4.7) is done exactly as in Step 1. If m is large enough so that t (Nm 1) T ν, then v m (Nm 1) = v m (Nm) = and (4.1a) is satisfied. We can modify v m for the remaining m (for example by setting v m = ) to ensure that this property holds for any m. By definition (4.3) of Π (θ) v m at t =, u(x, ) Π (θ) v m (x, ) = ϕ()w(x) ϕ(t () )Π Dm (P Dm w)(x) = ϕ()[w(x) Π Dm (P Dm w)(x)]. Then, by definition of P Dm and Ŝ, v() Π (θ) v m () = ϕ() w Π Dm (P Dm w) L2 () ϕ() Ŝ (w) L2 () and (4.1b) follows from the space time-consistency of ((D T ) m ) m N. To establish (4.1c), we define Ψ m : (, T ] R as the piecewise constant function equal to ϕ(t(n+1) ) ϕ(t (n) ) on (t (n), t (n+1) ], for all n =,..., N δt (n+ 2 1 m 1. ) Then, by definition (4.4) of δ Dm v m, for all t (, T ) and a.e. x, t v(x, t) δ Dm v m (x, t) = ϕ (t)w(x) Ψ m (t)π Dm (P Dm w)(x). By smoothness of ϕ we have Ψ m ϕ uniformly on [, T ] as m, and the proof of (4.1c) therefore follows by using the same sequence of estimates as in Step 1, starting from (4.13) and replacing (v, Π Dm v m, ϕ, Φ m ) with ( t v, δ Dm v m, ϕ, Ψ m ), and the L p norms with L 2 norms (note that Ŝ Dm (w) w Π Dm (P Dm w) L 2 () by definition of ŜD). 4.2 Averaged-in-time compactness Abstract setting In this section, we gather a few notions and compactness results used in the convergence analysis of numerical methods for time-dependent problems. The first two theorems are generalisations to vector-valued Lebesgue spaces of the classical Kolmogorov compactness theorem for L p spaces [13]. If E is a measured space and B a Banach space, we denote by L p (E; B) the Lebesgue space of p-integrable functions E B, see e.g. [31, 47] for a definition and some properties of these spaces.

113 12 4 Time-dependent problems: GDM and DFA Theorem 4.1 (Kolmogorov (1)). Let B be a Banach space, 1 p < +, T > and A L p (, T ; B). Then A is relatively compact in L p (, T ; B) if it satisfies the following conditions: 1. For all f A, there exists P f L p (R; B) such that P f = f a.e. on (, T ) and P f Lp (R;B) C, where C depends only on A. 2. For all ϕ Cc (R), the set { (P f)ϕdt, f A} is relatively compact in R B. 3. P f( + h) P f L p (R;B) as h, uniformly with respect to f A. Remark 4.11 (Necessary conditions) The conditions 1, 2 and 3 are actually also necessary for A to be relatively compact in L p (, T ; B). Proof. Let (ρ m ) m N be a sequence of mollifiers constructed by scaling a given smooth function ρ, that is: ρ Cc ( 1, 1), ρdt = 1, ρ, ρ( t) = ρ(t) for all t R R (4.16) and, for all m N and t R, ρ m (t) = mρ(mt). We set K = [, T ] and, for m N, A m = {(P f ρ m ) K, f A} where denotes the convolution product in R. The proof is divided in two steps. In Step 1 we prove, using Ascoli s theorem and Assumption 2, that, for m N, the set A m is relatively compact in C(K; B) endowed with its usual topology of the supremum norm. This easily gives the relative compactness of A m in L p (, T ; B). In Step 2, we show that Assumptions 1 and 3 give P f ρ m P f in L p (R; B) as m +, uniformly with respect to f A. This allows to conclude that the set A is relatively compact in L p (, T ; B). Step 1. Let m N. In order to prove that A m is relatively compact in C(K; B), we use Ascoli Arzelà s theorem. Hence, we need to prove that: (AA1) for all t K, the set {P f ρ m (t), f A} is relatively compact in B; (AA2) the sequence {P f ρ m, f A} is equicontinuous from K to B (i.e. the continuity of P f ρ m : K B is uniform with respect to f A). We first prove Property (AA1). For t K we have, with ϕ t = ρ m (t ) Cc (R), P f ρ m (t) = P f(s)ρ m (t s)ds = P f(s)ϕ t (s)ds. R Then, Assumption 2 applied to ϕ = ϕ t gives Property (AA1). We now prove Property (AA2). Let t 1, t 2 K and recall that p = Hölder s inequality, R p p 1. By

114 4.2 Averaged-in-time compactness 13 P f ρ m (t 2 ) P f ρ m (t 1 ) B P f(s) B ρ m (t 2 s) ρ m (t 1 s) ds R P f L p (R;B) ρ m(t 2 ) ρ m (t 1 ) L p (R). Since t 1, t 2 K = [, T ], the functions ρ m (t 2 ) and ρ m (t 1 ) vanish outside [ 1, T + 1]. Hence, using the mean value theorem and Assumption 3, we infer ( ) P f ρ m (t 2 ) P f ρ m (t 1 ) B C t 1 t 2 sup ρ m(t) (T + 2) 1 p. t R This shows that P f ρ m is uniformly continuous on R, with a modulus of continuity which does not depend on f. Hence, Property (AA2) is proved. As a consequence, A m is indeed relatively compact in C(K; B). This is equivalent to saying that, for any ε >, there exists a finite number of balls of radius ε (for the supremum norm of C(K; B)) whose union cover the set A m. Then, since L p (,T ;B) T 1/p C(K;B), we also obtain the relative compactness of A m in L p (, T ; B). Step 2. Let t R, we have, using R ρ m(s)ds = 1 and setting s = ms, P f ρ m (t) P f(t) = [P f(t s) P f(t)] ρ m (s)ds = R 1 1 [ ( P f t s ) ] P f(t) ρ(s)ds. m Then, by Hölder s inequality, 1 P f ρ m (t) P f(t) p B ρ p L p (t P f s ) P f(t) m Integrating with respect to t R and using the Fubini-Tonelli theorem to swap the integrals on t and s leads to 1 p B ds. P f ρ m P f p L p (,T ;B) 1 ρ p L p ( P f s ) P f m 1 p L p (,T ;B) 2 ρ p L p sup { P f( + h) P f p L p (R;B) : h 1 m Using Assumption 3 then gives P f ρ m P f L p (,T ;B) as m +, uniformly with respect to f A. We can now conclude the proof. Let ε > and pick m(ε) large enough such that ds }.

115 14 4 Time-dependent problems: GDM and DFA P f ρm(ε) P f Lp ε/2 for all f A. (4.17) (,T ;B) By Step 1, we can cover A m(ε) = {(P f ρ m(ε) ) [,T ] : f A} by a finite number of balls in L p (, T ; B) of radius ε/2. Property (4.17) then shows that {(P f) [,T ] : f A} = A is covered by the same finite number of balls of radius ε. This concludes the proof that A is relatively compact in L p (, T ; B). Theorem 4.12 (Kolmogorov (2)). Let B be a Banach space, 1 p < +, T > and A L p (, T ; B). Then A is relatively compact in L p (, T ; B) if it satisfies the following conditions: 1. A is bounded in L p (, T ; B). 2. For all ϕ Cc (R), the set { T fϕdt : f A} is relatively compact in B. 3. There exists a function η : (, T ) [, ) such that lim h + η(h) = and, for all h (, T ) and f A, T h f(t + h) f(t) p B dt η(h). Proof. The proof uses Theorem 4.1 with P defined the following way: for f A, P f = f on [, T ] and P f = on R\[, T ]. Owing to this definition and to Assumption 1 in Theorem 4.12, Items 1 and 2 of Theorem 4.1 are clearly satisfied. We prove now prove, in two steps, Item 3 of Theorem 4.1. Notice first that, replacing η with η(h) = sup (,h] η (which still satisfies lim h + η(h) = ), we can assume without loss of generality that η is non-decreasing. Step 1. In this step, we prove that τ f(t) p B dt as τ +, uniformly with respect to f A. Let τ, h (, T ) such that τ + h T. For all t (, τ) one has f(t) B f(t + h) B + f(t + h) f(t) B and thus, by the power-of-sums inequality (C.12), f(t) p B 2p 1 f(t + h) p B + 2p 1 f(t + h) f(t) p B. Integrating this inequality for t (, τ) gives τ τ f(t) p B dt 2p 1 f(t + h) p B dt + 2 p 1 τ f(t + h) f(t) p B dt. (4.18) Now let h (, T ) et τ (, T h ). For all h (, h ), Inequality (4.18) gives, using η(h) η(h ),

116 τ 4.2 Averaged-in-time compactness 15 τ f(t) p B dt 2p 1 f(t + h) p B dt + 2p 1 η(h ). Integrating this inequality over h (, h ) leads to τ h ( τ ) h f(t) p B dt 2p 1 f(t + h) p B dt dh+2 p 1 h η(h ). (4.19) Using the Fubini-Tonelli Theorem, h ( τ ) f(t + h) p B dt dh = τ τ ( ) h f(t + h) p B dh dt ( ) T f(s) p B ds τ f p L p (,T ;B), from which one deduces, owing to (4.19), τ f(t) p τ2p 1 B dt f p L h p (,T ;B) + 2p 1 η(h ). We can now conclude this step. Let ε > and choose h (, T ) such that 2 p 1 η(h ) ε. Then, with C = sup f A f p L p (,T ;B), take τ = min(t h, εh /(2 p 1 C)). This gives, for all f A and all τ τ, τ f(t) p B 2ε. The proof that τ f(t) p B dt as τ +, uniformly with respect to f A, is complete. A similar proof gives T T τ f(t) p B dt as τ +, uniformly with respect to f A (this can for example be obtained by working on g(t) = f(t t) instead of f). Step 2. We now prove that Item 3 in Theorem 4.1 is satisfied, and thus conclude the proof of Theorem Recall that P f(t) = if t [, T ] so that, for all h (, T ) and f A, P f(t + h) P f(t) p B dt R h h T h T f(t + h) p B dt + f(t + h) f(t) p B dt + f(t) p B dt + η(h) + T T h T h f(t) p B dt f(t) p B dt. (4.2) Let ε > and take h 1 > such η(h 1 ) ε. Owing to Step 1, there exists h 2 > such that, for all f A and h h 2,

117 16 4 Time-dependent problems: GDM and DFA h f(t) p B dt ε and T T h f(t) p B dt ε. Hence, by (4.2), for all f A and h min(h 1, h 2 ), P f(t + h) P f(t) p B dt 3ε. R This concludes the proof that Assumption 3 in Theorem 4.1 is satisfied. We now turn to compactness theorems involving sequences of spaces as codomains of the functions. This typically occurs in numerical schemes, when we consider sequences of functions that are piecewise constant on varying meshes. We first state a notion of compact embedding of a sequence of space in a fixed Banach space. Definition 4.13 (Compactly embedded sequence). Let B be a Banach space and (X m, Xm ) m N be a sequence of Banach spaces included in B. We say that the sequence (X m ) m N is compactly embedded in B if any sequence (u m ) m N such that u m X m for all m N, and ( u m Xm ) m N is bounded, is relatively compact in B. The first compactness results for sequences of subspaces is a straightforward translation in that setting of the second Kolmogorov theorem above. Proposition 4.14 (Time compactness with a sequence of subspaces). Let 1 p < +, T >, B be a Banach space, and (X m ) m N be compactly embedded in B (see Definition 4.13). Let (f m ) m N be a sequence in L p (, T ; B) satisfying the following conditions: 1. The sequence (f m ) m N is bounded in L p (, T ; B). 2. The sequence ( f m L 1 (,T ;X m) ) m N is bounded. 3. There exists a function η : (, T ) [, ) such that lim h + η(h) = and, for all h (, T ) and m N, T h f m (t + h) f m (t) p B dt η(h). Then, the sequence (f m ) m N is relatively compact in L p (, T ; B). Proof. We aim at applying Theorem 4.12 with A = {f m : m N}. We only have to prove Assumption 2 in this theorem, the other two assumptions being already stated as assumptions of the proposition. Let ϕ Cc (R). We need to prove that the sequence ( T f mϕdt) m N is relatively compact in B. We have, with ϕ = sup t R ϕ(t),

118 T 4.2 Averaged-in-time compactness 17 f m ϕdt ϕ f m L 1 (,T ;X. m) Xm The sequence ( f m L1 (,T ;X ) m) m N being bounded, this shows that the sequence ( ) T f m ϕdt Xm m N is also bounded. Since (X m ) m N is compactly embedded in B, this concludes the proof that ( T f mϕdt) m N in relatively compact in B. We then turn to the statement and proof of a discrete Aubin Simon theorem, which was first used in[6] and generalised in [59], see also [58]. In the continuous setting, the Aubin Simon compactness theorem establishes a strong compactness property of sequences of functions in L p (, T ; B), based on their boundedness in L q (, T ; A) and the boundedness of their derivatives in L r (, T ; C), where A is compactly embedded in B and B is continuously embedded in C. We first define a notion of triplets (A, B, C) having these compact continuous embedding properties, in the case where A and C are replaced by sequences of spaces. Definition 4.15 (Compactly continuously embedded sequence). Let B be a Banach space, (X m, Xm ) m N be a sequence of Banach spaces included in B, and (Y m, Ym ) m N be a sequence of Banach spaces. We say that the sequence (X m, Y m ) m N is compactly continuously embedded in B if the following conditions are satisfied: 1. The sequence (X m ) m N is compactly embedded in B (see Definition 4.13). 2. X m Y m for all m N and, for any sequence (u m ) m N such that a) u m X m for all m N and ( u m Xm ) m N is bounded, b) u m Ym as n +, c) (u m ) m N converges in B, it holds u m in B. Lemma Let B be a Banach space and (X m, Y m ) m N be compactly continuously embedded in B (see Definition 4.15). Then, for any ε >, there exists m N and C ε such that, for any m m and w X m, one has w B ε w Xm + C ε w Ym. Proof. We prove the result by contradiction. Let us therefore assume that there exists ε > such that, for any m N, we can find m = ϕ(m ) m and w X ϕ(m) such that w B > ε w Xϕ(m ) + m w Yϕ(m ). There is loss of generality in also selecting, by induction, each m = ϕ(m ) greater than ϕ(m 1); then ϕ : N N is a strictly increasing mapping. Since

119 18 4 Time-dependent problems: GDM and DFA w w B w, we can then set u ϕ(m) = X ϕ(m) (there is no ambiguity in the definition of u ϕ(m) since ϕ is one-to-one). We then have, for any m ϕ(n), 1 = u m B ε u m Xm + ψ(m) u m Ym, (4.21) where ψ = ϕ 1 : ϕ(n) N satisfies ψ(m) as m. To define u m for all m N, we let u m = whenever n ϕ(n) and, defining ψ(m) = m in that case, we see that (4.21) still holds. This definition also preserves the property ψ(m) as m. The sequence (u m ) m N is such that u m X m for all m N and, owing to (4.21), ( u m Xm ) m N is bounded by 1/ε. By the compact embedding of (X m ) m N in B, we infer that there exists a subsequence, still denoted (u m ) m N, that converges in B. Then, using (4.21) again, u m Ym 1/ψ(m) as m +, and thus, by Definition 4.15, the limit of (u m ) m N in B must be. This contradicts (4.21) which states that, since each u m has norm 1 in B, the limit in this space of these vectors should also have norm 1. We can now state a discrete Aubin Simon theorem with sequences of spaces. Theorem 4.17 (Aubin Simon with sequences of spaces and discrete derivative). Let p [1, + ). Let B be a Banach space and (X m, Y m ) m N be compactly continuously embedded in B (see Definition 4.15). Let T >, θ [, 1], and (f m ) m N be a sequence of L p (, T ; B) satisfying the following properties: 1. For all m N, there exists N N, = t () < t (1) < < t (N) = T, and (v (n) ) n=,...,n Xm N+1 such that, for all n {,..., N 1} and a.e. t (t (n), t (n+1) ), f m (t) = θv (n+1) + (1 θ)v (n). We then define almost everywhere the discrete derivative δ m f m by setting, with δt (n+ 1 2 ) = t (n+1) t (n), δ m f m (t) = v(n+1) v (n) δt (n+ 1 2 ) for n {,..., N 1} and t (t (n), t (n+1) ). 2. The sequence (f m ) m N is bounded in L p (, T ; B). 3. The sequence ( f m Lp (,T ;X m) ) m N is bounded. 4. The sequence ( δ m f m L 1 (,T ;Y m) ) m N is bounded. Then (f m ) m N is relatively compact in L p (, T ; B). Proof. We apply Proposition The only assumption in this proposition that needs to be established in order to conclude is the third one, that is T h f m (t + h) f m (t) p B dt as h, uniformly w.r.t. m N.

120 4.2 Averaged-in-time compactness 19 Note that, without the unformly with respect to m N, this convergence is known since each f m belongs to L p (, T ; B). As a consequence, we only have to prove that, for all η >, there exist m N and < h < T such that m m, h (, h ), T h f m ( + h) f m p B dt η. (4.22) Indeed, once this is proved, upon reducing h we can ensure that this estimate also holds for f 1,..., f m 1. Let ε >. Lemma 4.16 gives the existence of m N and C ε R such that, for all m m and u X m, u B ε u Xm + C ε u Ym. Then, for m m, < h < T and t (, T h), f m (t + h) f m (t) B ε f m (t + h) f m (t) Xm + C ε f m (t + h) f m (t) Ym ε f m (t + h) Xm + ε f m (t) Xm + C ε f m (t + h) f m (t) Ym. Take the power p of this inequality and use the power-of-sums inequality (C.14) to obtain f m (t + h) f m (t) p B 3p 1 ε p f m (t + h) p X m + 3 p 1 ε p f m (t) p X m + 3 p 1 C p ε f m (t + h) f m (t) p Y m. Integrating this inequality with respect to t (, T h) leads to T h f m (t + h) f m (t) p B dt 2 3p 1 ε p f m p L p (,T ;X m) + 3 p 1 C p ε T h f m (t + h) f m (t) p Y m dt. (4.23) We now estimate the last term in this inequality by using the discrete derivative of f m. This function is piecewise constant in time so, for a.e. t (, T h), writing f m (t + h) f m (t) as the sum of the jumps of f m at its discontinuities gives f m (t + h) f m (t) = (f m ) (t (n),t (n+1) ) (f m ) (t (n 1),t (n) ) = = = n: t (n) (t,t+h) n: t (n) (t,t+h) N 1 n=1 n: t (n) (t,t+h) (θv (n+1) + (1 θ)v (n) ) (θv (n) + (1 θ)v (n 1) ) [ ] θ(v (n+1) v (n) ) + (1 θ)(v (n) v (n 1) ) [ ] θ(v (n+1) v (n) ) + (1 θ)(v (n) v (n 1) ) 1 (t,t+h) (t (n) )

121 11 4 Time-dependent problems: GDM and DFA N 1 = θ n=1 v (n+1) v (n) δt (n+ 1 2 ) δt (n+ 1 2 ) 1 (t,t+h) (t (n) ) N 1 + (1 θ) n=1 v (n) v (n 1) δt (n 1 2 ) δt (n 1 2 ) 1 (t,t+h) (t (n) ), (4.24) where 1 (t,t+h) (t (n) ) = 1 if t (n) (t, t + h) and 1 (t,t+h) (t (n) ) = if t (n) (t, t + h). Let M be a bound of δ m f m L 1 (,T ;Y m), which means that, for all m N, N 1 v (n+1) v (n) δt (n+ 1 2 ) δt (n+ 1 2 ) M. Ym n= Taking the Y m -norm of (4.24), then the power p, and using the convexity of s s p gives f m (t + h) f m (t) p Y m ( N 1 ) p θ v (n+1) v (n) n=1 δt (n+ 1 2 ) δt (n+ 1 2 ) 1 (t,t+h) (t (n) ) Ym ( N 1 ) p + (1 θ) v (n) v (n 1) n=1 δt (n 1 2 ) δt (n 1 2 ) 1 (t,t+h) (t (n) ) Ym ( N 1 ) θm p 1 v (n+1) v (n) n=1 δt (n+ 1 2 ) δt (n+ 1 2 ) 1 (t,t+h) (t (n) ) Ym ( N 1 ) + (1 θ)m p 1 v (n) v (n 1) δt (n 1 2 ) δt (n 1 2 ) 1 (t,t+h) (t (n) ). (4.25) Ym n=1 Writing 1 (t,t+h) (t (n) ) = 1 (t (n) h,t (n) )(t) and integrating this inequality over t (, T h) leads to T h Plugging this inequality into (4.23), we obtain T h f m (t + h) f m (t) p Y m dt M p h. (4.26) f m (t + h) f m (t) p B dt 2 3p 1 ε p f m p L p (,T ;X m) + 3 p 1 C p ε M p h. (4.27) We can now conclude the proof. Let η >. Since ( f m Lp (,T ;X m) ) m N is bounded, we can fix ε (and thus also m ) such that, for all m m, 2 3 p 1 ε p f m p L p (,T ;X m) η 2. We can then select h (, T ) such that 3 p 1 C p ε M p h η/2. Estimate (4.27) then shows that (4.22) holds, which proves the theorem.

122 4.2 Averaged-in-time compactness Application to space time gradient discretisations In this section, we apply the previous abstract compactness results to the framework of space time GD. Aubin Simon theorem The first step is to defined a dual norm w,d on Π D (X D, ), which will enable us to define the spaces Y m in the above theorems. Definition 4.18 (Dual norm on Π D (X D, )). Let D T be a space time GD in the sense of Definition 4.1. The dual norm,d on Π D (X D, ) L 2 () is defined by: w Π D (X D, ), { } w,d = sup w(x)π D v(x)dx : v X D,, v D = 1. (4.28) A straightforward consequence of this definition is w Π D (X D, ), v X D,, w(x)π D v(x)dx w,d v D. (4.29) This relation shows that,d is a norm (not just a semi-norm). Indeed, if w,d = then w(x)π Dv(x)dx = for all v X D,. Taking then v such that Π D v = w shows that w =. The norm,d will mostly be used on δ D v(t) for v X N+1 D,. Recalling the notation (4.4), it is clear that δ D v(t) Π D (X D, ) for a.e. t (, T ), and thus δ D v(t),d is well-defined. Remark 4.19 (Boundary conditions) It is also worth noticing that w,d takes into account the considered boundary conditions, through the norm v D on X D, (see, e.g., Definitions 2.1 and 2.24). Remark 4.2 (,D is a discrete H 1 norm) Let us consider the case of homogeneous Dirichlet boundary conditions and p = 2. Then Definition 2.1 shows that (X D,, D ) is a discrete version of (H 1 (), H 1 () ), where H 1 () = L 2 () d is the standard norm on H 1 (). In the continuous setting, (4.28) therefore reads: for w L 2 (), { } w := sup w(x)v(x)dx : v H 1 (), v H 1 () = 1. (4.3) Identifying w as an element of H 1 (), we have w(x)v(x)dx = w, v H 1,H 1.

123 112 4 Time-dependent problems: GDM and DFA Hence, (4.3) turns out to be the standard dual norm on H 1 (), that is the norm of linear continuous functions H 1 () R. The norm,d can thus be considered as a discrete version of the standard dual norm on H 1 (). The next result is a consequence of the discrete Aubin Simon theorem (Theorem 4.17). Theorem 4.21 (Aubin Simon theorem for GDs). Let T >, p (1, + ) and θ [, 1]. Assume that ((D T ) m ) m N is a sequence of space time-consistent and compact space time GDs in the sense of Definitions 4.3 and 4.6. For any m N, let v m X Nm+1, be such that there exists C > satisfying and m N, m N, T T (v m ) θ (t) p dt C (4.31) δ Dm v m (t),dm dt C. (4.32) Then the sequence (Π (θ) v m ) m N is relatively compact in L p ( (, T )). Proof. To apply Theorem 4.17, let B = L p (), X m = Π Dm (X Dm, ), and define the norm on X m by u Xm = min{ w Dm : w X Dm, such that Π Dm w = u}. (4.33) Set Y m = X m = Π Dm (X Dm, ) and Ym =,Dm. Let us prove that the sequence (X m, Y m ) m N is compactly continuously embedded in B, in the sense of Definition First, the compactness hypothesis on ( ) m N is exactly stating that (X m ) m N is compactly embedded in B, in the sense of Definition Then, by construction, X m = Y m for all m N. Assume now that (u m ) m N is such that u m X m for all m N, ( u m Xm ) m N bounded, u m Ym as m +, and (u m ) m N converges in L p (). Take R m u m X Dm, a lifting of u m with minimal norm, i.e. Π Dm R m u m = u m and R m u m Dm = u m Xm A use of (4.29) yields u m (x) 2 dx = u m (x)π Dm R m u m (x)dx u m,dm R m u m Dm = u m Ym u m Xm. The assumptions on (u m ) m N thus ensure that lim m u m(x) 2 dx =. This shows that, up to a subsequence, u m a.e. in, and hence that the limit L p () of (u m ) m N must be. The proof (X m, Y m ) m N is compactly continuously embedded in B = L p () is complete.

124 4.2 Averaged-in-time compactness 113 The relative compactness of (Π (θ) v m ) m N in L p (, T ; L p ()) follows from Theorem 4.17 with f m = Π (θ) v m if we can check the four assumptions stated in this theorem. The first of this assumption is obviously satisfied by the definition of Π (θ) in (4.2). Since the sequence of underlying spatial discretisations is compact, it is also coercive (see, e.g., Lemma 2.9 for homogeneous Dirichlet boundary conditions). The definition of C Dm combined with (4.31) and the definition of Π (θ) then shows that (Π (θ) v m ) m N is bounded in L p (, T ; L p ()). This takes care of the second assumption in Theorem The third assumption follows immediately from (4.31) and the fact that Π (θ) v m (t) = Π Dm ((v m ) θ (t)) Xm (v m ) θ (t) Dm. Xm To prove the fourth assumption in Theorem 4.17, we notice that δ m Π (θ) v m (t) = δ Dm v m (t) Ym = δ Dm v m (t),dm Ym and we use (4.32). Convergence of a weak strong product, and identification of non-linear weak limits Dealing with degenerate parabolic equations often requires fine results to identify non-linear limits of weakly converging sequences. The main result in this section, Theorem 4.24, is one of these fine results. We consider here the particular case p = 2 and we restrict ourselves to homogeneous Dirichlet boundary conditions. The adaptation to other boundary conditions is rather simple, but establishing the equivalent of Theorem 4.24 for p 2 requires a different path; see [35, Theorem 5.4] for details. We first define the inverses of the discrete and continuous Laplace operators with homogeneous Dirichlet boundary conditions. Definition 4.22 (Inverse of discrete and continuous Laplace operator). Let D = (X D,, Π D, D ) be a GD in the sense of Definition 2.1, for p = 2. We define the operator i D : X D, X D, such that, for all v X D,, w X D,, D ( i Dv)(x) D w(x)dx = Π D v(x)π D w(x)dx. (4.34) We also define i : L 2 () H 1 () such that, for all v L 2 (), w H 1 (), ( i v)(x) w(x)dx = v(x)w(x)dx. (4.35)

125 114 4 Time-dependent problems: GDM and DFA Theorem 4.23 (Compactness of i ). Let p = 2, T >, θ [, 1] and ((D T ) m ) m N = (, I Dm, (t (n) m ) n=,...,nm ) m N be a space time-consistent, limit-conforming and compact sequence of space time GDs for homogeneous Dirichlet boundary conditions, in the sense of Definitions 4.3 and 4.6. For any m N, let v m X Nm+1, be such that there exists C > and q 1 satisfying and m N, m N, T T Π (θ) v m (t) 2 dt C (4.36) L 2 () δ Dm v m (t) q, dt C. (4.37) We also assume that Π (θ) v m converges weakly in L 2 (, T ; L 2 ()) as m to some v L 2 (, T ; L 2 ()). Then, as m, Moreover, if q > 1 then Π (θ) ( i v m ) i v in L 2 (, T ; L 2 ()), and (θ) ( i v m ) ( i v) in L 2 (, T ; L 2 ()) d. (4.38) δ Dm ( i v m ) t ( i v) weakly in L q (, T ; L 2 ()) as m. (4.39) Proof. Step 1: we prove, using Proposition 4.14 with p = 2, that (Π (θ) ( i v m )) m N is relatively compact in L 2 (, T ; L 2 ()). Let u m = i v m. Since the sequence of underlying spatial discretisations is compact, it is coercive (Lemma 2.9). Denote by C P a coercivity constant of this sequence. Using the definition (4.34) of i with v = (v m ) θ (t) and w = (u m ) θ (t), the Cauchy Schwarz inequality in the right-hand side, the definition of C P C Dm, raising to the power 2 and integrating over t (, T ), we see that T (u m ) θ (t) 2 CP 2 T Π (θ) v m (t) 2 C L 2 P 2 C. (4.4) () Set B = L 2 () and X m = Π Dm (X Dm,), endowed with the norm w Xm = inf{ z Dm : Π Dm z = w}. The compactness of ( ) m N ensures that (X m ) m N is compactly embedded in B as per Definition Estimate (4.4) and the coercivity of ( ) m N prove that items 1 and 2 of the hypotheses of Proposition 4.14 hold for f m = Π (θ) u m. Let us now observe that, for all z X Dm,, Π Dm z,dm

126 4.2 Averaged-in-time compactness 115 { } = sup Dm ( i z)(x) Dm w(x)dx : w X Dm, w Dm = 1 = i Dm z Dm. (4.41) Therefore, since δ Dm u m (t) = Π Du (n+1) m Π D u (n) m δt (n+ 1 2 ) of coercivity with constant C P imply that, Hypothesis (4.37) and the use m N, T δ Dm u m (t) q L 2 () dt (C P ) q C. (4.42) Apply the same computation as in (4.24), followed by (4.25) with Y m replaced by L 2 (). Using (4.42), this proves that (4.26) holds (still with L 2 () instead of Y m ). Hence, item 3 of the hypotheses of Proposition 4.14 holds with η(h) = h. Note that, contrary to the proof of Theorem 4.17, we do not use an inequality similar to (4.23), which is the discrete equivalent of the Lions lemma. Therefore, Proposition 4.14 provides the existence of u L 2 (, T ; L 2 ()) such that, up to a subsequence as m, Π (θ) u m u in L 2 (, T ; L 2 ()). Step 2: we prove that u = i v. By Lemma 4.7, u belongs to L 2 (, T ; H 1 ()) and (θ) u m u weakly in L 2 (, T ; L 2 () d ). Let w L 2 (, T ; H 1 ()) and consider the sequence (w m ) m N given by Lemma 4.9 for w. Writing (4.34) with v = (v m ) θ (t) and w = (w m ) θ (t) and integrating over t (, T ), we can pass to the limit to see that u satisfies T T u(x, t) w(x, t)dxdt = v(x, t)w(x, t)dxdt. (4.43) This precisely shows that u = i v. Step 3: proof of (4.38). Write now (4.34) with v = (v m ) θ (t) and w = (u m ) θ (t), and integrate over t (, T ). By strong convergence of Π (θ) u m to u, we can pass to the limit in the left-hand side and, using (4.43) with w = u, we find T lim (θ) m u m (x, t) 2 dxdt T = v(x)u(x)dxdt = T u(x, t) 2 dxdt. This convergence of L 2 norms show that the convergence of ( (θ) u m ) m N to u is actually strong. The proof of (4.38) is thus complete. Step 4: assuming that q > 1, proof of (4.39).

127 116 4 Time-dependent problems: GDM and DFA We proved in Step 1 that ( δdm ( i v m ) Lq (,T ;Y m) ) m N is bounded (recall that i v m = u m ). By coercivity of the sequence of GDs, this shows that δ Dm ( i v m ) is bounded in L q (, T ; L 2 ()) and therefore converges, up to a subsequence, to some V weakly in this space. Take γ Cc (, T ) and ψ Cc (). Multiply δ Dm ( i v m )(t) by [νγ(t (n) + (1 ν)γ(t (n+1) )]ψ, where ν = 1 θ and n is such that t (t (n), t (n+1) ), integrate over (x, t) (, T ) and use the discrete integration-by-part formula (C.17) to transfer the δ Dm operator onto (γ(t (n) )) n=,...,n. By smoothness of γ, passing to the limit shows that T T V (x, t)γ(t)ψ(x)dxdt = u(x, t)γ (t)ψ(x)dxdt. We infer that V = t u = t ( i v) and the proof is complete. The next result is characterised as weak strong space time because it deals with the product of two sequences of functions, one of them being strongly compact in time and weakly in space (estimates on the time derivative), the other one being wearkly compact in time and strongly in space (estimate on the spatial derivatives). Theorem 4.24 (Weak-strong space time convergence of a product). Take T >, θ [, 1], p = 2 and a space time-consistent, limit-conforming and compact sequence ((D T ) m ) m N of space time GDs for homogeneous Dirichlet boundary conditions, in the sense of Definitions 4.3 and 4.6. For any m N, let β m, ζ m X Nm+1, be such that The sequences ( T δ β m (t),dm ) m N and ( (θ) ζ m L 2 (,T ;L 2 () )) d m N are bounded, As m, Π (θ) β m β and Π (θ) ζ m ζ weakly in L 2 ( (, T )). Then it holds T lim Π (θ) m β m (x, t)π (θ) ζ m (x, t)dxdt T = β(x, t) ζ(x, t)dxdt. (4.44) Proof. The sequence (β m ) m N satisfies the hypotheses of Theorem Hence, (θ) ( i β m ) converges strongly to ( i β) in L 2 (, T ; L 2 () d ). By definition of i, we have T Π (θ) β m (x, t)π (θ) ζ m (x, t)dxdt T = (θ) ( i β m )(x, t) (θ) ζ m (x, t)dxdt. (4.45)

128 4.3 Uniform-in-time compactness 117 By assumption on (ζ m ) m N, the sequence ( ζ m L 2 (,T ;X Dm, ) m N is bounded and thus, by Lemma 4.7, (θ) ζ m ζ weakly in L 2 (, T ; L 2 () d ). Passing to the limit in the right-hand side of (4.45), we infer T lim Π (θ) m β m (x, t)π (θ) ζ m (x, t)dxdt T = ( i β)(x, t) ζ(x, t)dxdt. The definition of i concludes the proof of (4.44). 4.3 Uniform-in-time compactness Most of the results and techniques developed here come from [35] Definitions and abstract results Solutions of numerical schemes for parabolic equations are usually piecewise constant in time, and therefore not continous. Their jumps nevertheless tend to become small with the time step, and it is possible to establish some uniform-in-time convergence results. These results are based on a generalisation to non-continuous functions of the Ascoli Arzelà theorem. Definition If (K, d K ) and (E, d E ) are metric spaces, we denote by F(K, E) the space of functions K E, endowed with the uniform metric d F (v, w) = sup s K d E (v(s), w(s)) (note that this metric may take infinite values). Theorem 4.26 (Discontinuous Ascoli Arzelà s theorem). Let (K, d K ) be a compact metric space, (E, d E ) be a complete metric space, and let (F(K, E), d F ) be as in Definition Let (v m ) m N be a sequence in F(K, E) such that there exists a function ω : K K [, ] and a sequence (τ m ) m N [, ) satisfying lim ω(s, d K (s,s ) s ) =, lim τ m =, (4.46) m (s, s ) K 2, m N, d E (v m (s), v m (s )) ω(s, s ) + τ m. (4.47) We also assume that, for all s K, {v m (s) : m N} is relatively compact in (E, d E ). Then (v m ) m N is relatively compact in (F(K, E), d F ), and any adherence value of (v m ) m N in this space is continuous K E.

129 118 4 Time-dependent problems: GDM and DFA Proof. The last conclusion of the theorem, i.e. that any adherence value v of (v m ) m N in F(K, E) is continuous, is trivially obtained by passing to the limit along this subsequence in (4.47), showing that the modulus of continuity of v is bounded above by ω. The proof of the compactness result an easy generalisation of the proof of the classical Ascoli Arzelà compactness result. We start by taking a countable dense subset {s l : l N} in K (the existence of this set is ensured since K is compact metric). Since each set {v m (s l ) : m N} is relatively compact in E, by diagonal extraction we can select a subsequence of (v m ) m N, denoted the same way, such that for any l N, (v m (s l )) m N converges in E. We then proceed in showing that (v m ) m N is a Cauchy sequence in (F(K, E), d F ). Since this space is complete, this will show that this sequence converges in this space and will therefore complete the proof. Let ε > and, using (4.46), take δ > and M N such that ω(s, s ) ε whenever d K (s, s ) δ and τ m ε whenever m M. Select a finite set {s l1,..., s ln } such that any s K is within distance δ of a s li. Then, for any m, m M, by (4.47), d E (v m (s), v m (s)) d E (v m (s), v m (s li )) + d E (v m (s li ), v m (s li )) + d E (v m (s li ), v m (s)) ω(s, s li ) + τ m + d E (v m (s li ), v m (s li )) + ω(s, s li ) + τ m 4ε + d E (v m (s li ), v m (s li )). (4.48) Let i {1,..., N}. The sequence (v m (s li )) m N converges in E, and is therefore a Cauchy sequence in this space. We can thus find M i N such that m, m M i, d E (v m (s li ), v m (s li )) ε. (4.49) Take M = max(m, M 1,..., M N ). Estimates (4.49) and (4.48) show that, for all m, m M and all s K, d E (v m (s), v m (s)) 5ε. This concludes the proof that (v m ) m N is a Cauchy sequence in (F(K, E), d F ). Corollary 4.27 (Uniform-in-time compactness from estimates on discrete derivatives). Let T >, θ [, 1], B be a Banach space, and (X m, Xm ) m N be a sequence of Banach spaces included in B. For any m N, we take N m N, = t () m < t (1) m <... < t (Nm) m u m = (u (n) m ) n=,...,nm X Nm+1 m. = T, and Let (u m ) θ : [, T ] X m be the piecewise-constant function in time defined by (u m ) θ () = u () m and, for all n =,..., N m 1 and t (t (n), t (n+1) ], (u m ) θ (t) = θu (n+1) m + (1 θ)u (n) m. (4.5)

130 4.3 Uniform-in-time compactness 119 Set δt (n+ 1 2 ) m = t (n+1) m derivative δ m u m by: t (n) m for n =,..., N m 1, and define the discrete n =,..., N m 1, We assume that for a.e. t (t (n) m, t (n+1) m ), δ m u m (t) = u(n+1) m u (n) m δt (n+ 1 2 ) m (h1) The sequence (X m ) m N is compactly embedded in B (see Definition 4.13). (h2) The sequence ( (u m ) θ L (,T ;X m) ) m N is bounded. (h3) The sequence ( δ m u m Lq (,T ;B) ) m N is bounded for some q > 1. (h4) Setting δt m = max n=,...,nm 1 δt (n+ 1 2 ) m, it holds lim m δt m =. Then, there exists u C([, T ]; B) such that, up to a subsequence, Proof. Let u (n+θ) m = θu (n+1) m lim sup m t [,T ] (u m ) θ (t) u(t) B =. (4.51) + (1 θ)u (n) m and δ (n+ 1 2 ) m u m := u(n+1) m u (n) m δt (n+ 1 2 ) m Take n 2 n 1 in {,..., N m 1}, s 1 (t (n1), t (n1+1) ] and s 2 (t (n2), t (n2+1) ]. By writing a telescopic sum, we get (u m ) θ (s 2 ) (u m ) θ (s 1 ) = u (n2+θ) = = = m n 2 n=n 1+1 n 2 n=n 1+1 n 2 n=n 1+1 u (n1+θ) m [ u (n+θ) m ] u (n 1+θ) m [ ] θ(u (n+1) m u (n) m ) + (1 θ)(u m (n) u (n 1) m ) [ ] θδt (n+ 1 2 ) m δ (n+ 1 2 ) m u m + (1 θ)δt (n 1 2 ) m δ (n 1 2 ) m u m. (4.52) It can easily be checked that this relation extends to the case s 1 =, n 1 = 1 and n 2 {,..., N m 1} by defining δt ( 1 2 ) m = and δ ( 1 2 ) m u m = ; consider for example n 2 = and notice that u (θ) m u () m = θ(u (1) m u () m ) = θδt ( 1 2 ) m δ ( 1 2 ) m u m. By the discrete Hölder inequality (C.3) with ω i = δt (i± 1 2 ) m, b i = 1 and a i = δ (i± 1 2 ) m u m B, since q q = q 1,..

131 12 4 Time-dependent problems: GDM and DFA ( n 2 n=n 1+1 δt (n± 1 2 ) m ( n2 n=n 1+1 ) δ (n+ 1 2 ) q B u m m δt (n± 1 2 ) m ) q 1 ( n2 n=n 1+1 δt (n± 1 2 ) m ) δ (n± 1 2 ) q u m [ ] t (n ± 1 2 ) m t (n ± 1 2 ) q 1 m C q, (4.53) where C is a bound of ( δ m u m Lq (,T ;B) ) m N. Take the norm in B of (4.52), use the triangle inequality, then take the power q and use the convexity of s s q. Invoking finally the estimate (4.53) yields m B (u m ) θ (s 2 ) (u m ) θ (s 1 ) q B θc q [ t (n2+1) m t (n1+1) m ] q 1 + (1 θ)c q [ t (n2) m ] q 1 t m (n1), where we set t ( 1) m =. This gives C 1, that depends only on C and q, such that, for all s 1, s 2 [, T ] and all m N, (u m ) θ (s 1 ) (u m ) θ (s 2 ) B C 1 ( s 2 s 1 + δt m ) q 1 q C 1 s 2 s 1 q 1 q q 1 q + C 1 δt m. (4.54) In the last line, we used the power-of-sums inequality (C.13). This relation and (h4) show that v m = (u m ) θ satisfies Assumptions (4.46) (4.47) in the discontinuous Ascoli Arzelà theorem (Theorem 4.26), with K = [, T ] and E = B. The proof of Corollary 4.27 is therefore complete if we can establish that, for all s [, T ], {(u m ) θ (s) : m N} is relatively compact in B. (4.55) Assume first that s >. Since (u m ) θ is piecewise constant on (, T ], the L (, T ; X m ) norm of (u m ) θ is actually a supremum norm on (, T ]. Hence, (u m ) θ (s) Xm (u m ) θ L (,T ;X m) and Hypotheses (h1) and (h2) show that ((u m ) θ (s)) m N is indeed relatively compact in B. Let us now consider the case s =. Since (4.55) holds for any s >, by diagonal extraction we can find a subsequence, still denoted by (u m ) m N, such that, for any k N satisfying k 1 (, T ], the sequence ((u m ) θ (k 1 )) m N converges in B. We now prove that, along the same subsequence, ((u m ) θ ()) m N is a Cauchy sequence in B. This will conclude the proof that (4.55) holds for any s =. Owing to (4.54) we have, for (m, m ) N 2 and k N such that k 1 T, (u m ) θ () (u m ) θ () B (um ) θ () (u m ) θ (k 1 ) B + (um ) θ (k 1 ) (u m ) θ (k 1 ) B

132 4.3 Uniform-in-time compactness (um ) θ (k 1 ) (u m ) θ () B 2C 1 k q 1 q q 1 q + C 1 δt m q 1 q + C 1 δt m + (um ) θ (k 1 ) (u m ) θ (k 1 ) B. Given ε >, fix k such that 2C 1 k q 1 q < ε/4. Using (h4) and the convergence of ((u m ) θ (k 1 )) m N, we can then find m = m (k) N such that, if m, m m, q 1 q C 1 δt m ε q 1 4, C q 1δt m ε 4 and (um ) θ (k 1 ) (u m ) θ (k 1 ) B ε 4. This shows that (u m ) θ () (u m ) θ () B ε whenever m, m m. The sequence ((u m ) θ ()) m N is therefore Cauchy in B, and the proof is complete. The following lemma states an equivalent condition for the uniform convergence of functions, which proves extremely useful to establish uniform-in-time convergence of numerical schemes for parabolic equations when no smoothness is assumed on the data. Lemma Let (K, d K ) be a compact metric space, (E, d E ) be a metric space and (F(K, E), d F ) be as in Definition Let (v m ) m N be a sequence in F(K, E), and let v F(K, E). The following properties are equivalent. 1. v C(K, E) and v m v for d F, 2. for any s K and for any sequence (s m ) m N K converging to s for d K, we have v m (s m ) v(s) for d E. Proof. Step 1: Property 1 implies Property 2. For any sequence (s m ) m N converging to s, d E (v m (s m ), v(s)) d E (v m (s m ), v(s m )) + d E (v(s m ), v(s)) d F (v m, v) + d E (v(s m ), v(s)). The right-hand side tends to by definition of v m v for d F, and by continuity of v. Step 2: Property 2 implies Property 1. Let us first prove that v C(K, E). Let (s m ) m N K be a sequence converging to s for d K. Since for any t K the sequence (v n (t)) n N converges to v(t), we can find ϕ() N such that d E (v ϕ() (s ), v(s )) < 1. Assuming that, for n N, ϕ(n 1) N is given, we can also find ϕ(n) N such that ϕ(n) > ϕ(n 1) and d E (v ϕ(n) (s n ), v(s n )) < 1/(n + 1). We define the sequence (ŝ m ) m N by ŝ m = s n if m = ϕ(n) for some n N, and ŝ m = s if m ϕ(n). The sequence (ŝ m ) m N is constructed by interlacing the sequence (s m ) m N and the constant sequence equal to s. Hence, ŝ m s as m and, by assumption, (v m (ŝ m )) m N converges to v(s). The

133 122 4 Time-dependent problems: GDM and DFA sequence (v ϕ(n) (s n )) n N is a subsequence of (v m (ŝ m )) m N, and it therefore also converges to v(s). A triangle inequality then gives d E (v(s n ), v(s)) d E (v(s n ), v ϕ(n) (s n )) + d E (v ϕ(n) (s n ), v(s)) 1 n d E(v ϕ(n) (s n ), v(s)), which shows that v(s n ) v(s). This completes the proof that v C(K, E). We now prove by way of contradiction that v m v for d F. If (v m ) m N does not converge to v for d F, then there exists ε > and a subsequence (v mk ) k N, such that, for any k N, sup s K d E (v mk (s), v(s)) ε. We can then find a sequence (r k ) k N K such that, for any k N, d E (v mk (r k ), v(r k )) ε/2. (4.56) K being compact, up to another subsequence, denoted the same way, we can assume that r k s in K as k. As before, we then construct a sequence (s m ) m N converging to s, such that s mk = r k for all k N and s m = s if m {r k : k N}. By assumption, v m (s m ) v(s) in E and, by continuity of v, v(s m ) v(s) in E. A triangle inequality then shows that d E (v m (s m ), v(s m )), which contradicts (4.56) and concludes the proof. Uniform-in-time convergence of numerical solutions to schemes for parabolic equations often starts with a weak convergence with respect to the time variable. This weak convergence is then used to prove a stronger convergence. The following definition and proposition recall standard notions related to the weak topology on L 2 (). The inner product in L 2 () is denoted by, L2 (). Definition 4.29 (Uniform-in-time L 2 ()-weak convergence). Let (u m ) m N and u be functions [, T ] L 2 (). We say that (u m ) m N converges weakly in L 2 () uniformly on [, T ] to u if, for all ϕ L 2 (), as m the sequence of functions t [, T ] u m (t), ϕ L2 () converges uniformly on [, T ] to the function t [, T ] u(t), ϕ L2 (). Proposition 4.3. Let E be a closed bounded ball in L 2 () and let {ϕ l : l N} be a dense set in L 2 (). Then, on E, the weak topology of L 2 () is given by the metric d E (v, w) = l N min(1, v w, ϕ l L2 () ) 2 l. (4.57) Moreover, a sequence of functions u m : [, T ] E converges uniformly to u : [, T ] E for the weak topology of L 2 () if and only if, as m, the sequence of functions d E (u m, u) : [, T ] [, ) converges uniformly to.

134 4.3 Uniform-in-time compactness 123 Proof. The sets E ϕ,ε = {v E : v, ϕ L 2 () < ε}, for ϕ L 2 () and ε >, define a neighborhood basis of for the L 2 ()-weak topology on E. A neighborhood basis of any other points is obtained by translation of this particular basis. If R is the radius of the ball E then, for any ϕ L 2 (), l N and v E, v, ϕ L 2 () R ϕ ϕ l L 2 () + v, ϕ l L 2 (). By density of {ϕ l : l N} we can select l N such that ϕ ϕ l L2 () ε/(2r), and we then see that E ϕl,ε/2 E ϕ,ε. Hence, a neighborhood basis of in E for the L 2 ()-weak topology is also given by (E ϕl,ε) l N, ε>. From the definition of d E we see that, for any l N, min(1, v, ϕ l L 2 () ) 2 l d E (, v). If d E (, v) < 2 l this shows that v, ϕ l L 2 () 2 l d E (, v) and therefore that B de (, min(2 l, ε2 l )) E ϕl,ε. Hence, any neighborhood of in E for the L 2 ()-weak topology is a neighborhood of for d E. Conversely, for any ε >, selecting N N such that l N+1 2 l < ε/2 gives, from the definition (4.57) of d E, N E ϕl,ε/4 B de (, ε). l=1 Hence, any ball for d E centred at is a neighborhood of for the L 2 ()-weak topology. Since d E and L 2 ()-weak neighborhoods are invariant by translation, this concludes the proof that this weak topology is identical to the topology generated by d E. The conclusion on weak uniform convergence of sequences of functions follows from the preceding result, and more precisely by noticing that all previous inclusions are, when applied to u m (t) u(t), uniform with respect to t [, T ] Application to space time gradient discretisations We now consider applications of the previous results to the framework of space time gradient discretisations for generic boundary conditions, as described in Section 4.1. The following theorem is a consequence of Corollary Theorem 4.31 (L (, T ; L p ()) compactness). Let p (1, + ), T >, θ [, 1], and ((D T ) m ) m N be a space time-consistent, limit-conforming and compact sequence of space time GDs in the sense of Definitions 4.3 and 4.6. For each m N, let v m X Nm+1,. Assume that there exist C > and q > 1 satisfying m N, (v m ) θ L (,T ;X Dm, ) C, (4.58)

135 124 4 Time-dependent problems: GDM and DFA and m N, δ Dm v m L q (,T ;L p ()) C. (4.59) Then, there exists u C([, T ]; L p ()) L (, T ; W 1,p ()) and a subsequence, again denoted by ((D T ) m, v m ) m N, such that lim sup Π (θ) v m (t) u(t) =. (4.6) Lp () m t [,T ] Moreover, t u L q (, T ; L p ()) and, along the same subsequence, δ Dm v m t u weakly in L q (, T ; L p ()). Proof. We apply Corollary 4.27 with B = L p (), X m = Π Dm (X Dm, ) endowed with the norm (4.33), and u m (n) = Π Dm v m (n). The compactness hypothesis on ( ) m N states that (X m ) m N is compactly embedded in B in the sense of Definition 4.13, which yields Hypothesis (h1) in Corollary Hypothesis (h2) is satisfied owing to (4.58) and (u m ) θ (t) Xm = Π Dm [(v m ) θ (t)] Xm (v m ) θ (t) Dm. Hypothesis (h3) of Corollary 4.27 is obtained by (4.59) since δ m u m = δ Dm v m. Hypothesis (h4) is included in the definition of space time-consistency of ((D T ) m ) m N (Definition 4.3). By Corollary 4.27, we obtain u C([, T ]; L p ()) such that, up to a subsequence, (4.6) holds. The fact that u belongs to L (, T ; W 1,p ()) follows by Lemma 4.7. It remains to prove the convergence of the discrete time derivative. By (4.59) we can assume, upon extraction of a new subsequence, that δ Dm v m U weakly in L q (, T ; L p ()). The proof is complete by showing that U = t u in the sense of distributions on (, T ) (this also proves in particular that no further extraction was necessary). Take ψ Cc ( (, T )) and write, by definition (4.4) of δ Dm v m, T N m 1 = n=1 δ Dm v m (x, t)ψ(x, t)dxdt 1 δt (n+ 1 2 ) Set ψ n (x) = 1 t (n+1) t (n) t (n+1) δt (n+ 1 2 ) ( Π Dm v (n+1) m ) (x) Π Dm v m (n) (x) ψ(x, t)dxdt. (4.61) ψ(x, t)dt and, for ν = 1 θ, ψ t (n) n+ν = νψ n+1 + (1 ν)ψ n. Since ψ is smooth, ψ n (x) ψ n+ν (x) C ψ δt Dm for some C ψ not depending on x or n. Using the discrete integration-by-parts formula (C.17), (4.61) yields, for m large enough so that ψ = ψ Nm = (which is possible due to ψ vanishing on a neighbourhood of and T ),

136 T = = N m 1 n=1 N m 1 = = 4.3 Uniform-in-time compactness 125 δ Dm v m (x, t)ψ(x, t)dxdt (4.62) n=1 N m 1 ( ( n=1 N m 1 t (n+1) n=1 Π Dm v (n+1) m Π Dm v (n+1) m t (n) where, owing to (4.59), ) (x) Π Dm v m (n) (x) ψ n (x)dx ) (x) Π Dm v m (n) (x) ψ n+ν (x)dx + R m Π Dm v m (n+θ) (x) (ψ n+1 (x) ψ n (x)) dx + R m Π Dm v m (n+θ) (x) ψ n+1(x) ψ n (x) dx + R δt (n+ 1 2 ) m (4.63) R m C ψ δt Dm δ Dm v m L 1 ( (,T )) as m. The term (4.62) converges to T U(x, t)ψ(x, t)dxdt (4.64) and, owing to the smoothness of ψ and the convergence of Π (θ) v m, the term (4.63) converges to T u(x, t) t ψ(x, t)dxdt. (4.65) The proof that U = t u is complete by equating (4.64) and (4.65). The uniform-in-time weak-in-space compactness result provided by the next theorem is the initial step to proving a uniform-in-time strong-in-space convergence result for gradient scheme approximations of parabolic equations. Theorem 4.32 (Uniform-in-time L 2 ()-weak compactness). Let T >, θ [, 1] and ((D T ) m ) m N be a sequence of space time-consistent space time GDs in the sense of Definition 4.3. For each m N, let v m X Nm+1 D. m, Assume that there exists C > and q > 1 such that, for all m N, T sup Π (θ) v m (t) C and δ Dm v m (t) q,d L2 () m dt C (4.66) t [,T ] (see Definition 4.18 of,d ). Then, the sequence (Π (θ) v m ) m N is relatively compact weakly in L 2 () uniformly on [, T ], that is, it has a subsequence which converges accordingly to Definition Moreover, any limit of such a subsequence is continuous [, T ] L 2 () for the weak topology.

137 126 4 Time-dependent problems: GDM and DFA Proof. Theorem 4.32 is a consequence of the discontinuous Ascoli Arzelà theorem (Theorem 4.26), with K = [, T ] and E the ball of radius C in L 2 (), endowed with the weak topology. Let {ϕ l : l N} Cc () be a dense set in L 2 () and endow E with the metric (4.57) from these ϕ l. By Proposition 4.3, this metric defines the weak L 2 () topology. The set E is metric compact and therefore complete, and the functions Π (θ) v m have values in E. It remains to estimate d E (Π (θ) v m (s), Π (θ) v m (s )). We drop the index m in D for legibility. Let s s T and take n 1, n 2 {,..., N 1} such that s (t (n1), t (n1+1) ] and s (t (n2), t (n2+1) ]. If s = we let n 1 = 1 and t ( 1) =. In a similar way as (4.52), we write Π (θ) D v m(s ) Π (θ) v m(s) = θ n 2 n=n 1+1 D δt (n+ 1 2 ) δ (n+ 1 2 ) D v m + (1 θ) n=n 1+1 n 2 n=n 1+1 δt (n 1 2 ) δ (n 1 2 ) D v m, where δt ( 1 2 ) = and δ ( 1 2 ) D v m =. Take P D ϕ l X D, that realises the minimum defining S D (ϕ l ), multiply the previous relation by by Π D P D ϕ l and integrate over. Estimates (4.29), (4.66) and the Hölder inequality (C.3) (used as in (4.53)) yield ( ) Π (θ) D v m(x, s ) Π (θ) D v m(x, s) Π D P D ϕ l (x)dx n2 θ δt (n+ 1 2 ) δ (n+ 1 2 ) D v m (x)π D P D ϕ l (x)dx n=n 1+1 n2 + (1 θ) δt (n 1 2 ) δ (n 1 2 ) D v m (x)π D P D ϕ l (x)dx n 2 θ P D ϕ l D n=n 1+1 n 2 + (1 θ) P D ϕ l D δt (n+ 1 2 ) δ (n+ 1 2 ),D D v m n=n 1+1 C 1/q[ θ(t (n2+1) t (n1+1) ) 1/q δt (n 1 2 ) δ (n 1 2 ),D D v m + (1 θ)(t (n2) t (n1) ) 1/q ] P D ϕ l D. (4.67) By definition of P D and of D (depending on the specific boundary conditions), we have Π D P D ϕ l ϕ l L 2 () ŜD(ϕ l ) and, using a triangle inequality,

138 4.3 Uniform-in-time compactness 127 P D ϕ l D ŜD(ϕ l ) + D ϕl C ϕl where D ϕl and C ϕl do not depend on D (and therefore on m). Since t (n2+1) t (n1+1) s s + δt and t (n2) t (n1) s s + δt, the estimate on Π (θ) v m in (4.66) gives, owing to (4.67), ( ) Π (θ) D v m(x, s ) Π (θ) D v m(x, s) ϕ l (x)dx ( ) Π (θ) D v m(x, s ) Π (θ) D v m(x, s) Π D P D ϕ l (x)dx + 2CŜD(ϕ l ) C 1/q C ϕl s s 1/q + C 1/q C ϕl δt 1/q + 2CŜD(ϕ l ). (4.68) Plugged into the definition (4.57) of the distance in E, this yields ( ) d E Π (θ) D v m(s ), Π (θ) D v m(s) l N + l N min(1, C 1/q C ϕl s s 1/q ) 2 l min(1, 2CŜ (ϕ l ) + C 1/q C ϕl δt 1/q m ) 2 l =: ω(s, s ) + τ m. Using the dominated convergence theorem for series, we see that ω(s, s ) as s s, and that τ m as m (we invoke the space timeconsistency of ((D T ) m ) m N to see that lim m Ŝ Dm (ϕ l ) for any l). Hence, the assumptions of Theorem 4.26 are satisfied and the proof is complete. The following easy lemma is used in the proofs of uniform-in-time convergence and strong convergence of the gradient for gradient schemes approximations of parabolic equations. Lemma Let (a m ) m N and (b m ) m N be two sequences of real numbers, and a, b R. We assume that a lim inf m a m, b lim inf m b m and lim sup m (a m + b m ) a + b. Then a m a and b m b as m. Proof. We have a + b lim inf m a m + lim inf m b m lim inf (a m + b m ) lim sup(a m + b m ) a + b. m m Hence, all inequalities involved in this sequence are equalities and, in particular, lim inf m (a m + b m ) = lim sup m (a m + b m ) = lim m (a m + b m ) = a + b, and a = lim inf m a m. We then write lim sup m b m = lim sup(b m + a m a m ) m

139 128 4 Time-dependent problems: GDM and DFA lim sup(b m + a m ) + lim sup( a m ) m m = lim (a m + b m ) lim inf a m = a + b a = b. m m Combined with b lim inf m b m, this proves that b m b as m. We then have a m = a m + b m b m a + b b = a and the proof is complete.

140 5 Parabolic problems In this chapter, we consider time-dependent problems and their approximation by the gradient discretisation method (GDM). First, in Section 5.1, we study a quasi-linear problem, which is the transient version of the quasi-linear problem studied in Chapter 3. We first prove an error estimate for the GDM approximation of the linear version of this problem, under additional regularity hypotheses. For the complete non-linear problem, the mathematical arguments used in the convergence analysis of the GDM come from Chapter 4. The convergence of the gradient schemes (GS) for this problem is proved under minimal regularity on the solution. In Section 5.2, we analyse the convergence of the GDM applied to a nonconservative parabolic equation, which indludes the regularised level-set equations. For this model, additional regularity on the initial condition must be assumed. Finally, in Section 5.3, we turn to generalised (non-local) fully non-linear Leray Lions parabolic problems with Neumann boundary conditions. These problems arise in particular from image processing models. Several convergence results for the GDM are obtained, including a uniform-in-time strongin-space convergence result (based on the tools developed in Section 4.3). We stress that such a convergence implies in particular the pointwise-in-time convergence, which is of high practical interest. Indeed, users of numerical techniques are often more interested in approximating a quantity of interest at a given time, rather than averaged over a time span. 5.1 The gradient discretisation method for a quasilinear parabolic problem In the whole section, we let p = 2.

141 13 5 Parabolic problems The continuous problem We consider the following problem: approximate the solution u of t u div (Λ(x, u) u) = f + div(f ), in (, T ) with initial condition u(, ) = u ini, on, and homogeneous Dirichlet boundary conditions u = on (, T ). (5.1a) (5.1b) (5.1c) The following hypotheses are assumed: is an open bounded connected subset of R d, d N, and T >, Λ : (, T ) M d (R) is a Caratheodory function (i.e. Λ(x, s) is measurable w.r.t. x and continuous w.r.t. s), and there exists λ, λ > such that, for a.e. x, for all s R, Λ(x, s) is symmetric with eigenvalues in [λ, λ], f L 2 ( (, T )), F L 2 ( (, T )) d, u ini L 2 (). (5.2a) (5.2b) (5.2c) (5.2d) Under Hypotheses (5.2), a function u is a weak solution of (5.1) if u L 2 (, T ; H 1 ()) and, for all v L 2 (, T ; H 1 ()) such that t v L 2 ( (, T )) and v(, T ) =, T u(x, t) t v(x, t)dxdt u ini (x)v(x, )dx T + Λ(x, u(x, t)) u(x, t) v(x, t)dxdt = T (f(x, t)v(x, t) F (x, t) v(x, t))dxdt. (5.3) Taking v Cc ( (, T )) in this equation shows that (5.1a) then holds in the sense of distributions. Since Λ(x, u) u and F both belong to L 2 ( (, T )) d, this implies that t u L 2 (, T ; H 1 ()). As a consequence, u C([, T ]; L 2 ()) and, integrating by parts the first term in (5.3) and using the density of Cc ([, T ]; H 1 ()) in L 2 (, T ; H 1 ()) (see [31, Corollary 1.3.1]), we see that u satisfies

142 5.1 The gradient discretisation method for a quasilinear parabolic problem 131 u L 2 (, T ; H 1 ()) C([, T ]; L 2 ()), t u L 2 (, T ; H 1 ()), u(, ) = u ini and, for all w L 2 (, T ; H 1 ()), T t u(, t), w(, t) H 1,H 1dt T (5.4) + Λ(x, u(x, t)) u(x, t) w(x, t)dxdt T = (f(x, t)w(x, t) F (x, t) w(x, t))dxdt. Remark 5.1. The existence of at least one solution u to (5.3), and therefore to (5.4), will be a consequence of the convergence analysis of the GDM (see Remark 5.6). In the linear case, that is Λ(x, u) = Λ(x), estimates on the continuous solution show that this solution u is also unique The gradient scheme Recalling that p = 2, let D T = (X D,, Π D, D, I D, (t (n) ) n=,...,n ) and θ [ 1 2, 1] be a space time GD for homogeneous Dirichlet boundary conditions in the sense of Definition 4.1. Using a θ-scheme for the time stepping, the GDM applied to Problem (5.4) leads to the following GS: find a family (u (n) ) n=,...,n X N+1 D, such that, recalling the notations (4.2) and (4.4), u () = I D u ini and, for all n =,..., N 1, u (n+1) satisfies δ (n+ 1 2 ) D u(x)π D v(x)dx + Λ(x, Π D u (n+θ) (x)) D u (n+θ) (x) D v(x)dx (5.5) 1 t (n+1) = (f(x, t)π δt (n+ 1 2 ) D v(x) F (x, t) D v(x))dxdt, t (n) v X D,. Here, of course, u (n) is expected to provide an approximation of u at time t n. Remark 5.2 (Practical implementation of the GS (5.5)) For any n =,..., N 1, taking u (n+θ) as unknown, and using u (n+1) = u(n+θ) (1 θ)u (n), θ the implementation of the GS (5.5) is similar to that of the GS (3.44) for the steady quasilinear problem.

143 132 5 Parabolic problems Error estimate in the linear case We now consider Problem (5.1) under Hypotheses (5.2) and the following additional hypotheses. F = and Λ(, s) = Id. (5.6) The equation we consider is therefore t u u = f, with homogeneous Dirichlet boundary conditions. Theorem 5.3 (Error estimate, linear case and regular solution). Under Hypotheses (5.2) and (5.6), let D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. We assume the existence of h D > such that ϕ W 2, () H 1 (), S D (ϕ) h D ϕ W 2, (), ϕ W 1, () d, W D (ϕ) h D ϕ W 1, () d, ϕ W 1, () H 1 (), Π D I D ϕ ϕ L2 () h D ϕ W 1, (). (5.7a) (5.7b) (5.7c) Assume that the solution u to (5.4) belongs to W 1, (, T ; W 2, ()), and let u be the solution to the GS (5.5) with θ = 1. Then there exists C >, depending only on u,, T and (in a non-decreasing way) of C D, such that max Π (1) D u(, t) u(, t) L C(δt D + h D ) 2 () t [,T ] and (1) D u u L2 ( (,T )) d C(δt D + h D ). Remark 5.4 (Existence of h D) If I D is linear and continuous (such as, e.g., in Remark 4.4), there always exists h D satisfying (5.7). Indeed, defining P (2) D : H1 () X D, as in (5.8) below, the property (5.7) holds with h D upper bound of the norms of the following linear or bilinear operators: W 2, () H 1 () L 2 () L 2 () d, u (u Π DP (2) D (2) u, u DP u), D and W 1, () d X D, R, (ϕ, v) (divϕ(x)π Dv(x) + ϕ(x) Dv(x))dx, W 1, () H 1 () L 2 (), u u Π DI Du.

144 5.1 The gradient discretisation method for a quasilinear parabolic problem 133 Proof. In the following proof, we denote by C i various quantities having the same dependencies as C in the theorem. For the sake of brevity, if n {,..., N 1} and g = f, u or t u we set We also let u () = u(). g (n+1) 1 (x) = δt (n+ 1 2 ) t (n+1) t (n) g(x, t)dt. Step 1: a linear spatial interpolator. The interpolator P D defined by (3.1) enables us to plug the exact solution into the scheme, which is an essential process in establishing error estimates. However, this P D is not necessarily linear, which becomes a problem for parabolic equations. We therefore need a slightly modified version of this interpolator. We define P (2) D : H1 () X D, by: for ϕ H 1 (), ( ) P (2) D ϕ = argmin Π D w ϕ 2 L 2 () + Dw ϕ 2 L 2 (). (5.8) d w X D, Let V = {(Π D w, D w) : w X D, } and P : L 2 () L 2 () d V be the orthogonal projection. Since D L2 () is a norm on X D,, for any d z V there exists a unique Rz X D, such that (Π D Rz, D Rz) = z. This defines a linear continuous mapping R : V X D,, and (5.8) shows that P (2) D ϕ = R P(ϕ, ϕ) for all ϕ H1 (). Hence, P (2) D ϕ is uniquely defined and P (2) D is linear continuous. The characterisation of the orthogonal projection P also shows that, for all ϕ H 1 () and w X D,, Π D P (2) D ϕ(x)π Dw(x) + D P (2) D ϕ(x) Dw(x)dx = ϕ(x)π D w(x) + ϕ(x) D w(x)dx. Let ϕ H 1 (). Taking v X D, that realises the minimum defining S D (ϕ) and using the definition of P (2) D shows that ( ΠD P (2) D ϕ ϕ 2 + D P (2) L 2 D ϕ ϕ 2 ) 1/2 () L 2 () d ( ) 1/2 Π D v ϕ 2 L 2 () + Dv ϕ 2 L 2 () d 2SD (ϕ). (5.9) We have u C([, T ]; W 2, ()) and u W 1, (, T ; L 2 () d ), which implies that u : [, T ] L 2 () is Lipschitz-continuous. Hence, (5.9) with ϕ = u(t (n+1) ) and Assumption (5.7a) yield u (n+1) D P (2) u (n+1) u(t (n+1) ) L 2 () d D u(t(n+1) ) L 2 () d

145 134 5 Parabolic problems + S D (u(t (n+1) )) C 1 (δt D + h D ). (5.1) Since t u L (, T ; W 2, ()), the quantity t u (n+1) W is bounded 2, () independently of n. Applying (5.9) to ϕ = t u (n+1) = u(t(n+1) ) u(t (n) ), using δt (n+ 1 2 ) the linearity of P (2) D and invoking (5.7a), we obtain Π D P (2) D u(t(n+1) ) Π D P (2) D u(t(n) ) δt (n+ 1 2 ) t u (n+1) L2 () Step 2: proof of the error estimates. Since u (n+1) H div () we can write, for all v X D,, ( ) Π D v(x)div( u (n+1) )(x) + u (n+1) (x) D v(x) dx C 2 h D. (5.11) W D ( u (n+1) ) v D. Owing to the regularity of u, the equation t u f = div( u) is satisfied a.e. in space and time. Averaging over time in (t (n), t (n+1) ) gives t u (n+1) f (n+1) = div( u (n+1) ) a.e. in space, and thus ( ( ) ) Π D v(x) t u (n+1) (x) f (n+1) (x) + u (n+1) (x) D v(x) dx W D ( u (n+1) ) v D. Use the GS (5.5) to replace the term f (n+1) in the left-hand side. Since u L (, T ; W 1, () d ), the quantity u (n+1) W 1, () is bounded independently on n and Assumption (5.7b) yields d ( ) Π D v(x) t u (n+1) (x) δ (n+ 1 2 ) D u(x) dx ( + u (n+1) (x) D u (n+1)) D v(x)dx C 3 h D v D. (5.12) For k =,..., N, set e (k) = P (2) D and δ (n+ 1 2 ) D e = D e (n+1) = u(t(k) ) u (k). We have [ Π D P (2) D u(t(n+1) ) Π D P (2) D u(t(n) ) δt (n+ 1 2 ) [ + t u (n+1) δ (n+ 1 2 ) D ] u. t u (n+1) [ D P (2) D u(t(n+1) ) u (n+1)] + [ u (n+1) D u (n+1)] ]

146 5.1 The gradient discretisation method for a quasilinear parabolic problem 135 Then (5.12), (5.11), (5.1) and the definition of C D give Π D v(x)δ (n+ 1 2 ) D e(x)dx + D e (n+1) (x) D v(x)dx C 4 (δt D + h D ) v D. Take v = δt (n+ 1 2 ) e (n+1), and sum on n =,..., m 1 for some m {1,..., N}. Recalling the definition of D, m 1 n= [ ] Π D e (n+1) (x) Π D e (n+1) (x) Π D e (n) (x) dx δt (n+ 1 2 ) m 1 + n= N 1 n= We now apply the relation D e (n+1) (x) 2 dx C 4 (δt D + h D )δt (n+ 1 2 ) ( D e (n+1) (x) 2 dx) 1/2. (5.13) a, b R, b(b a) = 1 2 b2 1 2 a (b a)2 1 2 b2 1 2 a2 (5.14) to a = Π D e (n) (x) and b = Π D e (n+1) (x). Using the Young inequality (C.8) with p = p = 2 in the right-hand side of (5.13), this leads to N 1 1 (Π D e (m) (x)) 2 dx + δt (n+ 1 2 ) 2 n= (Π D e () (x)) 2 dx N 1 n= m 1 n= D e (n+1) (x) 2 dx δt (n+ 1 2 ) D e (n+1) (x) 2 dx C 2 4(δt D + h D ) 2 δt (n+ 1 2 ). (5.15) By Assumption (5.7c) and Estimate (5.9), since u () = I D u ini = I D u(), Π D e () L2 Π D P (2) D u() u() L2 + u() Π D I D u() L () () 2 () C 5 h D. Hence, recalling the definition of (1) and using N 1 n= δt(n+ 1 2 ) = T, Equation (5.15) yields 1 D e 2 (Π (m) (x)) 2 dx+ 1 2 t (m) (1) D e(x, t) 2 dxdt C 6 (δt D +h D ) 2. (5.16)

147 136 5 Parabolic problems Using a triangle inequality, (5.9), and the power-of-sums inequality (C.13) with α = 1/2, Equation (5.16) leads on one hand to m = 1,..., N, Π D u (m) u(t (m) ) C 7 (δt D + h D ) + 2S D (u(t (m) )) L 2 () C 8 (δt D + h D ). (5.17) On the other hand, using again (5.9) and a triangle inequality, Equations (5.16) with m = N 1 and the power-of-sums inequality (C.12) with α = 2 lead to N 1 δt (n+ 1 2 ) D u (n+1) u(t (n+1) ) n= 2 L 2 () N 1 4C 6 (δt D + h D ) δt (n+ 1 2 ) S D (u(t (n+1) )) 2 C9(δt 2 D + h D ) 2. (5.18) n= The conclusion follows from (5.17), (5.18) and the Lipschitz-continuity of u : [, T ] H 1 () to compare u(t) (resp. u(t)) with u(t (n+1) ) (resp. u(t (n+1) )) when t (t (n), t (n+1) ] Convergence analysis in the non-linear case We come back to the generic quasilinear model (5.1). The convergence result we intend on proving is the following. Theorem 5.5 (Convergence of the GDM). Under Assumptions (5.2), let θ [ 1 2, 1] and ((D T ) m ) m N be a sequence of space time GDs for homogeneous Dirichlet boundary conditions in the sense of Definition 4.1, which is space time-consistent, limit-conforming and compact in the sense of Definitions 4.3 and 4.6. For any m N, let u m be a solution to (5.5) with D T = (D T ) m. Then, up to a subsequence as m, sup Π (θ) u m (t) u(t) (5.19a) L2 () t [,T ] (θ) u m u in L 2 ( (, T )) d, (5.19b) where u is a solution to (5.3) (and thus also (5.4)). Remark 5.6. We do not assume the existence of a solution u to the continuous problem. The convergence analysis establishes this existence. The analysis of any GDM for non-linear models starts by establishing a priori estimates on the solution to the GS. These estimates first allow us to prove that such a solution exist, and are then useful to invoke the compactness results of Chapter 4.

148 5.1 The gradient discretisation method for a quasilinear parabolic problem 137 Lemma 5.7 (L (, T ; L 2 ()) estimate and discrete L 2 (, T ; H 1 ()) estimate). Under Assumptions (5.2), let θ [ 1 2, 1] and D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Let u be a solution to the corresponding GS (5.5). Then, for any k =,..., N, (Π D u (k) (x)) 2 dx t (k) sup Π (θ) t [,T ] Λ(x, Π (θ) D t (k) u(x, t)) (θ) (Π D I D u ini (x)) 2 dx (f(x, t)π (θ) D D u(x, t) (θ) D u(x, t)dxdt u(x, t) F (x, t) (θ) u(x, t))dxdt. D (5.2) Consequently, there exists C 1 > depending only on C P C D (see Definition 2.2), C ini Π D I D u ini L 2 (), f, F, and λ such that D u(t) L C 1 and (θ) 2 D u L C 1. (5.21) () 2 ( (,T )) d Moreover, there exists at least one solution u to the GS (5.5). Proof. Relation (5.14) is generalised to the following: for all a, b R, [( (a b)(θa + (1 θ)b) = (a b) θ 1 ) ( ) ] 1 a θ b + 1 (a b)(a + b) 2 ( = θ 1 ) (a b) (a2 b 2 ) 1 2 (a2 b 2 ). Let n {,..., N 1}. Applying the above relation to a = Π D u (n+1) and b = Π D u (n) yields δt (n+ 1 2 ) δ (n+ 1 2 ) D u Π D u (n+θ) 1 ( (Π D u (n+1) ) 2 (Π D u (n) ) 2). (5.22) 2 Setting v = δt (n+ 1 2 ) u (n+θ) in (5.5) and summing over n =,..., k 1 (we assume here that k 1, the case k = in (5.2) is trivial) therefore leads to 1 2 k 1 ( n= k 1 + (Π D u (n+1) (x)) 2 dx δt (n+ 1 2 ) n= k 1 t (n+1) n= t (n) ) (Π D u (n) (x)) 2 dx Λ(x, Π D u (n+θ) (x)) D u (n+θ) (x) D u (n+θ) (x)dx (f(x, t)π D u (n+θ) (x) F (x, t) D u (n+θ) (x))dxdt. (5.23)

149 138 5 Parabolic problems The first sum is telescopic and reduces to (Π D u (k) (x)) 2 dx (Π D u () (x)) 2 dx = (Π D u (k) (x)) 2 dx (Π D I D u ini (x)) 2 dx. Recalling that Π (θ) D u(x, t) = Π Du (n+θ) (x) and (θ) D u(x, t) = Du (n+θ) (x) whenever t (t (n), t (n+1) ], Equation (5.23) can then be recast as (5.2). Using the Cauchy Schwarz inequality (i.e. (C.5) with p = p = 2), the Young inequality (C.9) and the definition (2.1) of C D, we write t (k) (f(x, t)π (θ) D u(x, t) F (x, t) (θ) D u(x, t))dxdt f L2 ( (,t (k) )) Π (θ) D u L2 ( (,t (k) )) + F L 2 ( (,t (k) )) d (θ) D u L 2 ( (,t (k) )) d C2 D λ f 2 L 2 ( (,t (k) )) + C2 D λ 4C 2 D + 1 λ F 2 L 2 ( (,t (k) )) d + λ 4 Π (θ) D u 2 (θ) D u 2 L 2 ( (,t (k) )) L 2 ( (,t (k) )) d λ f 2 L 2 ( (,t (k) )) + 1 λ F 2 L 2 ( (,t (k) )) d + λ (θ) D 2 u 2. (5.24) L 2 ( (,t (k) )) d Plugged into (5.2) and using the coercivity of Λ, this gives 1 (Π D u (k) (x)) 2 dx + λ (θ) D 2 u 2 L 2 ( (,t (k) )) d Hence, 1 2 max k=,...,n 1 2 C2 ini + C2 D λ f 2 L 2 ( (,t (k) )) + 1 λ F 2 L 2 ( (,t (k) )) d + λ (θ) D 2 u 2. L 2 ( (,t (k) )) d Π D u (k) 2 L 2 () + λ 2 (θ) D u 2 L 2 ( (,T ) d 1 2 C2 ini + C2 D λ f 2 L 2 ( (,T )) + 1 λ F 2 L 2 ( (,T )) d. The estimates in (5.21) follow from this inequality and from the fact that, by definition (4.2) of Π (θ) D,

150 5.1 The gradient discretisation method for a quasilinear parabolic problem 139 Π D u (n+θ) L 2 () θ Π D u (n+1) L 2 () + (1 θ) Π D u (n) L 2 (). Following the same arguments as in the proof of Theorem 3.16, it is easy to establish by induction that, for each n =,..., N 1, there is a solution u (n+1) to the equation in (5.5). This shows that this GS has at least one solution u. Lemma 5.8 (Estimate on the dual norm of the discrete time derivative). Under Assumptions (5.2), let θ [ 1 2, 1] and D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Let u be a solution to the corresponding GS (5.5). Then there exists C 11, depending only on C P C D, C ini Π D I D u ini L 2 (), f, F, λ and λ, such that T where the dual norm,d is defined by (4.28). δ D u(t) 2,D dt C 11, (5.25) Proof. In (5.5), choose v X D, which realises the supremum in the definition (4.28) of δ (n+ 1 2 ) D u,d. Recalling that v D = 1 and applying the Cauchy Schwarz inequality as well as the definition (2.1) of C D, we get δ (n+ 1 2 ) D u λ D u (n+θ) L,D 2 () 1 + δt (n+ 1 2 ) 1 = δt (n+ 1 2 ) t (n+1) t (n) t (n+1) t (n) [ λ (C D f(, t) L 2 () + F (, t) L 2 () d)dt (θ) D u(t) L2 () + C D f(, t) L 2 () + F (, t) L2 () d ]dt Square, use the Jensen inequality (C.1), multiply by δt (n+ 1 2 ) and apply the power-of-sums inequality (C.14). Recalling the definition (4.4) of δ D u, this yields t (n+1) t (n) δ D u(t) 2,D dt t (n+1) [ ] 3 λ 2 (θ) D u(t) 2 + CP 2 f(, t) 2 L t (n) D 2 () + F (, t) 2 L 2 () dt. d We conclude the proof of (5.25) by summing over n =,..., N 1 and by invoking Estimates (5.21). We are now ready to prove the convergence of the GS (5.5). Proof of Theorem 5.5.

151 14 5 Parabolic problems We note that since ((D T ) m ) m N is compact, it is also coercive (see Lemma 2.9). Step 1: Application of compactness results. By Estimates (5.21), Lemma 4.7 gives the existence of some u L 2 (, T ; H 1 ()) such that, up to a subsequence as m, Π (θ) u m u weakly in L 2 ( (, T )) and (θ) u m u weakly in L 2 ( (, T )) d. Estimate (5.25) and Theorem 4.21 show that, in fact, Π (θ) u m converges strongly to u in L 2 ( (, T )). Step 2: u is a solution to (5.3) (and thus also (5.4)). Let v L 2 (, T ; H 1 ()) be such that t v L 2 ( (, T )) and v(t, ) =. Let (v m ) m N be given for v by Lemma 4.9 (with 1 θ instead of θ). In the following, we drop the index m in, N m and v m for legibility reasons. Introduce v (n+(1 θ)) as test function in (5.5), multiply by δt (n+ 1 2 ), and sum the result on n =,..., N 1. Recalling the definitions (4.2), this gives T (m) 1 + T (m) 2 = T (m) 3 with and T (m) 1 = N 1 n= T T (m) 2 = T ( T (m) 3 = [ ] Π D u (n+1) (x) Π D u (n) (x) Π D v (n+(1 θ)) (x)dx, Λ(x, Π (θ) D f(x, t)π (1 θ) D u(x, t)) (θ) D u(x, t) (1 θ) D v(x, t)dxdt, ) v(x, t) F (x, t) (1 θ) D v(x, t) dxdt. Applying the discrete integration-by-parts (C.17), with ν = 1 θ, to T (m) 1 and using the fact that v (N) =, we write T (m) 1 = N 1 n= = N 1 t (n+1) n= T = [ ] Π D u (n+θ) (x) Π D v (n+1) (x) Π D v (n) (x) dx Π D u () (x)π D v () (x)dx t (n) Π D u () (x)π D v () (x)dx Π (θ) D u(x, t)δ Dv(x, t)dxdt Π (θ) D u(x, t)δ Dv(x, t)dxdt Π D I D u ini (x)π (1 θ) D v(x, )dx.

152 5.1 The gradient discretisation method for a quasilinear parabolic problem 141 Recall that Π (θ) D u u in L2 ( (, T )) and, by space time-consistency of ((D T ) m ) m N, that Π D I D u ini u ini in L 2 (). The convergence properties of (v m ) m N stated in (4.1c) and (4.1b) (with 1 θ instead of θ) show that T lim T (m) m 1 = u(x, t) t v(x, t)dxdt u ini (x)v(x, )dx. (5.26) Since Π (θ) D u u in L2 ( (, T )), Lemma C.4 (non-linear strong convergence property) shows that Λ(, Π (θ) D u) (1 θ) D v converges to Λ(, u) v in L 2 ( (, T )) d as m. Hence, using the symmetry of Λ and the weak-strong convergence result of Lemma C.3, lim T (m) m 2 = lim = m T T The convergences of Π (1 θ) D T lim T (m) m 3 = (θ) D u(x, t) Λ(x, Π(θ) D u(x, t)) (1 θ) D v(x, t)dxdt Λ(x, u(x, t)) u(x, t) v(x, t)dxdt. (5.27) v and (1 θ) D v readily give (f(x, t)v(x, t) F (x, t) v(x, t)) dxdt. (5.28) Using (5.26), (5.27) and (5.28) to pass to the limit m in T (m) 1 + T (m) 2 = T (m) 3 shows that u satisfies the equation in (5.3). Step 3: Uniform-in-time convergence of Π (θ) u m. Let s [, T ] and (s m ) m 1 be a sequence in [, T ] that converges to s. Assume first that s m > and let k(m) {,..., N m 1} be such that s m (t (k(m)), t (k(m)+1) ]. By convexity of the square function and by Definition (4.2) of Π (θ) D, ( (Π (θ) u m (x, s m )) 2 = θπ Dm u (k(m)+1) m ) 2 + (1 θ)π Dm u (k(m)) m θ(π Dm u (k(m)+1) m ) 2 + (1 θ)(π Dm u (k(m)) m ) 2. (5.29) Set s ( ) m := t (k(m)) and s (+) m := t (k(m)+1), which both converge to s as m. Write (5.2) for k = k(m) + 1, multiply by θ, write (5.2) with k = k(m), and multiply by 1 θ. Summing the two inequalities thus obtained and using (5.29) yields

153 142 5 Parabolic problems 1 2 (Π (θ) u(x, s m )) 2 dx + [ + s ( ) m θ 1 2 s (+) m + s ( ) m + θ Λ(x, Π (θ) u(x, t)) (θ) u(x, t) (θ) u(x, t)dxdt s ( ) m Λ(x, Π (θ) u(x, t)) (θ) u(x, t) (θ) u(x, t)dxdt (Π Dm I Dm u ini (x)) 2 dx s (+) m s ( ) m (f(x, t)π (θ) u(x, t) F (x, t) (θ) u(x, t))dxdt (f(x, t)π (θ) u(x, t) F (x, t) (θ) u(x, t))dxdt. ] (5.3) Inequality (5.3) also obviously holds if s m = (with, in this case, s (+) m = s ( ) m = ). Our aim is to take the superior limit of (5.3). We first analyse the behaviour of all the terms, except the first one. The Cauchy Schwarz inequality for the semi-definite positive symmetric form W L 2 ( (, T )) d shows that ( s ( ) m ( s ( ) m s ( ) m Λ(x, Π (θ) u m (x, t))w (x, t) W (x, t)dxdt Λ(x, Π (θ) u m (x, t)) (θ) u m (x, t) u(x, t)dxdt ( s ( ) m As m, we have ) 2 Λ(x, Π (θ) u m (x, t)) (θ) u m (x, t) (θ) u m (x, t)dxdt Λ(x, Π (θ) u m (x, t)) u(x, t) u(x, t)dxdt Π (θ) u m u strongly in L 2 ( (, T )), and 1 [,s ( ) m ] u 1 [,s] u strongly in L 2 ( (, T )) d. ) ). (5.31) Here, 1 [a,b] is the function of time such that 1 [a,b] (t) = 1 if t [a, b], and 1 [a,b] (t) = otherwise. The non-linear strong convergence property stated in Lemma C.4 page 46 then shows that, as m, 1 ( ) [,s Λ(, Π(θ) m ] u m ) u 1 [,s] Λ(, u) u strongly in L 2 ( (, T )) d.

154 5.1 The gradient discretisation method for a quasilinear parabolic problem 143 Owing to Lemma C.3 (weak-strong convergence property) and to the weak convergence in L 2 ( (, T )) d of (θ) u m to u, the left-hand side of (5.31) and the second term in the right-hand side of (5.31) pass to the limit. Taking the inferior limit of this inequality and dividing by s Λ(x, u) u u, we deduce that s Λ(x, u(x, t)) u(x, t) u(x, t)dxdt lim inf m s ( ) m Λ(x, Π (θ) u m (x, t)) (θ) u m (x, t) (θ) u m (x, t)dxdt. (5.32) The space time-consistency of ((D T ) m ) m N (Definition 4.3) gives (Π Dm I Dm u ini (x)) 2 dx (u ini (x)) 2 dx as m. (5.33) Still considering m, we have 1 ( ) [,s m ] f 1 [,s]f in L 2 ( (, T )) and 1 ( ) [,s m ] F 1 [,s]f in L 2 ( (, T )) d. The weak convergences of Π (θ) u m and (θ) u m thus give, as m, s ( ) m Finally, since 1 [s ( ) L 2 ( (, T )) d, s (+) m s ( ) m (f(x, t)π (θ) u(x, t) F (x, t) (θ) u(x, t))dxdt s (f(x, t)u(x, t) F (x, t) u(x, t))dxdt. (5.34) m,s (+) m ] f in L2 ( (, T )) and 1 [s ( ) m,s (+) m ] F in (f(x, t)π (θ) u(x, t) F (x, t) (θ) u(x, t))dxdt. (5.35) We now come back to (5.3), drop the non-negative term in brackets, move the second term from the left-hand side to the right-hand side, and take the superior limit. The convergences (5.32), (5.33), (5.34) and (5.35) yield 1 lim sup (Π (θ) D m 2 m u m (x, s m )) 2 dx 1 s u ini (x) 2 dx + (f(x, t)u(x, t) F (x, t) u(x, t))dxdt 2 s ( ) m lim inf m 1 u ini (x) 2 dx + 2 Λ(x, Π (θ) u m (x, t)) (θ) u m (x, t) (θ) s u m (x, t)dxdt (f(x, t)u(x, t) F (x, t) u(x, t))dxdt

155 144 5 Parabolic problems s Λ(x, u(x, t)) u(x, t) u(x, t)dxdt. (5.36) Since u L 2 (, T ; H 1 ()) and t u L 2 (, T ; H 1 ()), the following integration by parts is justified (see [31, Section 2.5.2]): s t u(t), u(t) H 1,H 1dt = 1 u(x, s) 2 dx 1 u(x, ) 2 dx. 2 2 Making w = u1 [,s] (t) in (5.4), we therefore see that 1 s u(x, s) 2 dx + 2 = 1 u ini (x) 2 dx + 2 Λ(x, u(x, t)) u(x, t) u(x, t)dxdt (f(x, t)u(x, t) F (x, t) u(x, t))dxdt. s Used in (5.36), this relation gives lim sup (Π (θ) u m (x, s m )) 2 dx m (5.37) u(x, s) 2 dx, (5.38) Owing to Theorem 4.32 and to Estimates (5.21) and (5.25), (Π (θ) u m ) m N converges to u weakly in L 2 () uniformly in [, T ] (in the sense of Definition 4.29). Hence, Π (θ) u m (, s m ) u(, s m ) weakly in L 2 () as m. Estimate (5.38) and a standard reasoning in Hilbert spaces then show that this convergence is actually strong in L 2 (). By Lemma 4.28, we infer that (5.19a) holds. Step 4: Strong convergence of (θ) u m. Note that, by (5.19a), Π (θ) u m (, T ) u(, T ) in L 2 (). Write (5.3) with s m = T (so that s ( ) m = t (Nm 1) and s (+) m = T ), move the first term to the right-hand side and take the superior limit. We can pass, as in the previous step, to the limit in all the terms on the right-hand side. Let h m : [, T ] R be the function such that h m = 1 on [, s ( ) m ] and h m = θ on (s ( ) m, T ]. Using (5.37) with s = T, we obtain T lim sup h m (t)λ(x, Π (θ) u m (x, t)) (θ) u m (x, t) (θ) u m (x, t)dxdt m 1 T u ini (x) 2 dx + (f(x, t)u(x, t) F (x, t) u(x, t))dxdt 2 1 u(x, T ) 2 dx 2 T = Λ(x, u(x, t)) u(x, t) u(x, t)dxdt.

156 5.2 Non-conservative problems 145 Using this estimate, the strong convergence in L 2 ( (, T )) of Π (θ) u m to u, Lemma C.4, the strong convergence in L 2 ( (, T )) d of h m u to u, the weak convergence in L 2 ( (, T )) d of (θ) u m to u, and developing the following expression in a similar fashion as (3.52), we infer that T lim sup h m (t)λ(x, Π (θ) u m (x, t))( (θ) u m (x, t) u(x, t)) m ( (θ) u m (x, t) u(x, t))dxdt. By coercivity of Λ and since h m θ 1 2, this shows that, as m, T (θ) u m (x, t) u(x, t) 2 dxdt. This concludes the proof that (θ) u m u strongly in L 2 ( (, T )) d as m. Remark 5.9 (About the discrete IBP formula (C.17)) The usage in Step 2 of the ν -discrete integration by parts formula (C.17) is nonstandard. A usual way of proceeding, see e.g. [38] or the proof of Theorem 5.2, is to analyse in this proof the convergence of Π (1) u m towards u. Using this analysis, the test function v (n+1), instead of v (n+(1 θ)), can be used in Step 2, and the more standard discrete integration-by-parts formula (C.15) can then be applied. Thanks to (C.17), we can however fully analyse in the proof of Theorem 5.5 the convergence of Π (θ) u m without having to analyse at the same time Π (1) u m, which is a less natural reconstruction for θ-schemes. 5.2 Non-conservative problems The continuous problem We focus in this section on the approximation of some non-linear problems under the following non-conservative form: ν(x, t, u(x, t), u(x, t)) t u(x, t) div(µ( u(x, t) ) u(x, t)) = f(x, t), for a.e. (x, t) (, T ) (5.39a) with the initial condition u(x, ) = u ini (x), for a.e. x, (5.39b) and boundary conditions

157 146 5 Parabolic problems u(x, t) =, for a.e. (x, t) (, T ). (5.39c) The hypotheses are as follows: is an open bounded connected subset of R d (d N ) and T >, u ini H 1 () f L 2 ( (, T )), ν : (, T ) R R d R is a Caratheodory function and there exists ν max ν min > such that ν(x, t, s, ξ) [ν min, ν max ] for a.e. x, t and for all s, ξ, (5.4a) (5.4b) (5.4c) (5.4d) (Caratheodory means that, for all (s, ξ) R R d, the function (x, t) ν(x, t, s, ξ) is measurable and, for a.e. (x, t) (, T ), the function (s, ξ) ν(x, t, s, ξ) it is continuous) µ : R + R is Lipschitz-continuous, non-increasing, and there exists µ max µ min > and α > such that µ(s) [µ min, µ max ] and (sµ(s)) α for all s R +. (5.4e) On specific choice of µ and ν is of particular interest. For given real numbers < a b, using the functions ( 1 µ(s) = max s2 + a, 1 ), s R +,, 2 b ν(x, t, z, ξ) = µ( ξ ), (x, t) (, T ), z R, ξ R d in (5.39a) lead to to the regularised level set equation [48]. These functions satisfy (5.4d) (5.4e) with α = a 2 /b 3. Let us now give the precise mathematical meaning of a solution to Problem (5.39) under Hypotheses (5.4). Definition 5.1 (Weak solution of (5.39)). Under Hypotheses (5.4), we say that u is a weak solution of (5.39) if 1. u L 2 (, T ; H 1 ()) and t u L 2 ( (, T )) (which implies u C([, T ]; L 2 ())), 2. u(, ) = u ini, 3. the following holds

158 T + T ν(x, t, u(x, t), u(x, t)) t u(x, t)v(x, t)dxdt µ( u(x, t) ) u(x, t) v(x, t)dxdt T = f(x, t)v(x, t)dxdt, 5.2 Non-conservative problems 147 v L 2 (, T ; H 1 ()). (5.41) The third item shows that a weak solution to (5.39) satisfies (5.39a) in the sense of distributions. In particular, for such a solution, div(µ( u ) u) L 2 ( (, T )). Our aim is to use the GDM to construct gradient schemes for (5.41), and to prove their convergence to a weak solution of (5.39). As usual for non-linear model, convergence proofs start with a priori estimates. Let us formally show the kind of estimates that can be obtained on (5.39). Defining F by s s s R +, F (s) = zµ(z)dz [µ 2 min 2, µ s 2 ] max, (5.42) 2 any sufficiently regular function u satisfies d F ( u(x, t) )dx = µ( u(x, t) ) u(x, t) t u(x, t)dxdt. (5.43) dt Therefore, assuming that u is solution of (5.39a) with f = (for the sake of simplicity of this brief presentation) and taking v = t u in (5.41), we see that T ν(u, u) t u(x, t) 2 dxdt + F ( u(x, t) )dx = F ( u ini (x) )dx. (5.44) The discrete equivalent of this essential estimate is established in Lemma 5.11 for the fully-implicit scheme (using that x xµ(x) is strictly increasing), and in Lemma 5.15 for the semi-implicit scheme (using that µ is decreasing). The hypothesis that x xµ(x) is instrumental, for both schemes, to prove that the reconstructed gradients converge strongly Fully implicit scheme Let D T = (X D,, Π D, D, I D, (t (n) ) n=,...,n ) be a space time GD, for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1 with p = 2 and θ = 1. Using a fully implicit time-stepping, the GDM applied to Problem (5.41) leads to the following GS: find a family u = (u (n) ) n=,...,n X N+1 D, such that

159 148 5 Parabolic problems u () = I D u ini and, for n =,..., N 1, u (n+1) satisfies t (n+1) ν(x, t, Π D u (n+1), D u (n+1) )δ (n+ 1 2 ) D u(x)π D v(x)dxdt t (n) +δt (n+ 1 2 ) µ( D u (n+1) (x) ) D u (n+1) (x) D v(x)dx t (n+1) = f(x, t)π D v(x)dxdt, v X D,. t (n) (5.45) We recall the notations (4.2) and (4.4), and that θ = 1 here. The operators Π (1) D and (1) D will therefore be our natural space time function and gradient reconstructions. Estimates and existence of a solution to the fully implicit scheme Lemma 5.11 (L 2 ( (, T )) estimate on δ D u and L (, T ; X D, ) estimate on u, fully implicit scheme). Under Hypotheses (5.4), let D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Then, for any solution u to the GS (5.45) and for all m = 1,..., N, t (m) ν min δ D u(x, t) 2 D dxdt + µ min u (m) 2 L 2 () d µ max D I D u ini 2 L 2 () + 1 f 2 d L ν 2 ( (,T )). (5.46) min As a consequence, there exists at least one solution u to the GS (5.45). Proof. Setting v = u(n+1) u (n) leads to + t (m) ν min m 1 n= t (m) δt (n+ 1 2 ) in (5.45) and summing over n =,..., m 1 δ D u(x, t) 2 dxdt µ( D u (n+1) (x) ) D u (n+1) (x) [ ] D u (n+1) (x) D u (n) (x) dx f(x, t)δ D u(x, t)dxdt. (5.47) Hypothesis (5.4e) implies the convexity of F, defined by (5.42), and thus c 1, c 2 R +, F (c 2 ) F (c 1 ) = This gives in particular c2 c 1 zµ(z)dz c 2 µ(c 2 )(c 2 c 1 ).

160 5.2 Non-conservative problems 149 F ( D u (n+1) (x) ) F ( D u (n) (x) ) [ ] µ( D u (n+1) (x) ) D u (n+1) (x) D u (n+1) (x) D u (n) (x). (5.48) The Cauchy-Schwarz inequality implies [ D u (n+1) (x) ] D u (n+1) (x) D u (n) (x) D u (n+1) (x) [ D u (n+1) (x) D u (n) (x) Combining (5.48) and (5.49) and plugging the result into (5.47) yields t (m) ν min m 1 + n= t (m) δ D u(x, t) 2 dxdt [ ] F ( D u (n+1) (x) ) F ( D u (n) (x) ) dx ]. (5.49) f(x, t)δ D u(x, t)dxdt. (5.5) The sum in the left-hand side is telescopic and reduces to [ ] F ( D u (m) (x) ) F ( D u () (x) ) dx. The right-hand side of (5.5) can be estimated by means of the Cauchy Schwarz inequality and the Young inequality (C.9). Since the range of F is in [µ min s 2 /2, µ max s 2 /2], this gives t (m) ν min µ max 2 µ max 2 δ D u(x, t) 2 dxdt + µ min 2 D u (m) (x) 2 dx D u () (x) 2 dx + f L 2 ( (,T )) δ Du L 2 ( (,t (m) )) D I D u ini (x) 2 dx + 1 2ν min f 2 L 2 ( (,T )) + ν min 2 δ Du 2 L 2 ( (,t (m) )). Moving the last term to the left-hand side yields Estimate (5.46). To prove the existence of at least one solution to the GS, we create an homotopy between the model (5.39) and a linear PDE. By induction, it suffices to show that, for a given u (n) X D,, there exists u (n+1) X D, satisfying the integral relation in (5.45). For λ [, 1], define µ λ and ν λ by µ λ (s) = µ max (1 λ) + λµ(s), and ν λ (x, t, s, ξ) = ν min (1 λ) + λν(x, t, s, ξ).

161 15 5 Parabolic problems Let (v i ) i=1,...,m be a basis of X D, and define Φ : X D, [, 1] X D, by its components (Φ(w, λ) i ) i=1,...,m on (v i ) i=1,...,m : Φ(w, λ) i = t (n+1) t (n) + δt (n+ 1 2 ) t (n+1) t (n) ν λ (x, t, Π D w(x), D w(x)) Π Dw(x) Π D u (n) (x) Π δt (n+ 1 2 ) D v i (x)dxdt µ λ ( D w(x) ) D w(x) D v i (x)dx f(x, t)π D v i (x)dxdt. Then u (n+1) satisfies the integral equation in (5.45) if and only if Φ(u (n+1), 1) =. The mapping Φ is clearly continuous. If Φ(w, λ) = then, since µ λ (resp. ν λ ) has its range in [µ min, µ max ] (resp. [ν min, ν max ]), similar estimates to the ones established above give a bound on D w L 2 () = w d D that does not depend on λ [, 1]. Finally, for λ =, Φ(, ) is affine and therefore invertible since, by the previous bound, its kernel is bounded (and thus necessarily reduced to a single point). As a consequence, for some R large enough, Φ(, ) = has a solution in the ball of radius R in X D,. A topological degree argument (see Theorem C.1 page 45) can therefore be applied and show that Φ(, 1) = has at least one solution, i.e. that there exists u (n+1) solution to the integral equation in (5.45). Convergence of the fully implicit scheme For u X N+1 D,, define w D and G D by, for a.e. (x, t) (, T ), ( ) w D (x, t) = f(x, t) ν x, t, Π (1) D u(x, t), (1) D u(x, t) δ D u(x, t), (5.51) ( ) G D (x, t) = µ (1) D u(x, t) (1) D u(x, t). (5.52) With these definitions, (5.45) can be recast as u X N+1 D, and, for all v X N+1 D,, T G D (x, t) (1) D v(x, t)dxdt T = w D (x, t)π (1) D v(x, t)dxdt. (5.53) The following lemma is an initial step towards establishing the convergence of the fully implicity GS for (5.39).

162 5.2 Non-conservative problems 151 Lemma 5.12 (A convergence property of the fully implicit scheme). Under Hypotheses (5.4), let ((D T ) m ) m N be a sequence of space time GDs for homogeneous Dirichlet boundary conditions (with p = 2). Assume that the sequence ((D T ) m ) m N is space time-consistent, limit-conforming and compact in the sense of Definitions 4.3 and 4.6. Also assume that ( Dm I Dm u ini ) m N is bounded in L 2 () d. For any m N, take u m a solution to the GS (5.45) and define w Dm and from u m by (5.51) (5.52). Then there exist functions G Dm u L (, T ; H 1 ()) C([, T ]; L 2 ()) with t u L 2 ( (, T )) and u(, ) = u ini, G L 2 ( (, T )) d, and w L 2 ( (, T )) such that, along a subsequence as m, sup t [,T ] Π (1) u m (t) u(t) L2 (), (1) u m converges weakly in L 2 ( (, T )) d to u, δ Dm u m converges weakly in L 2 ( (, T )) to t u, G Dm converges weakly to G in L 2 ( (, T )) d, w Dm converges weakly to w in L 2 ( (, T )), it holds T G Dm (x, t) (1) u m (x, t)dxdt T G(x, t) u(x, t)dxdt. (5.54) Remark Note that, at this stage, we do not identify G and w, respectively, with µ( u ) u and ν(u, u). This is done later, in the proof of Theorem 5.14, by using (5.54). Proof. Owing to (5.46), (G Dm ) m N is bounded in L (, T ; L 2 () d ) and (w Dm ) m N is bounded in L 2 ( (, T )). Hence, there exists G L 2 ( (, T )) d and w L 2 ( (, T )) such that, up to a subsequence as m, G Dm G weakly in L 2 ( (, T )) d and w Dm w weakly in L 2 ( (, T )). By Lemma 5.11, the sequence ((u m ) 1 ) m N (see notation (4.2) with θ = 1) is bounded in L (, T ; X Dm,) and the sequence (δ Dm u m ) m N is bounded in L 2 (, T ; L 2 ()). Theorem 4.31 thus provides u L (, T ; H 1 ()) C([, T ]; L 2 ()) such that t u L 2 ( (, T )) and, up to a subsequence as m, sup t [,T ] Π (1) u m (t) u(t) L2 () and δ Dm u m t u weakly in L 2 ( (, T )). The weak convergence of (1) u m to u is a consequence of Lemma 4.7. The definition (4.3) gives Π (1) u m () = Π Dm u () m = Π Dm I Dm u ini. The space time-consistency of ((D T ) m ) m N then yields Π (1) u m () u ini in L 2 () as

163 152 5 Parabolic problems m. By the uniform convergence of (Π (1) u m ) m N to u, we infer that u(, ) = u ini. We now aim to prove (5.54). Since u L 2 (, T ; H 1 ()) we can take (v m ) m N given by Lemma 4.9 for v = u. Using v m as a test function in (5.53) with D T = (D T ) m and passing to the limit yields T G(x, t) u(x, t)dxdt = T w(x, t)u(x, t)dxdt. (5.55) Putting v = u m in (5.53), the weak-strong convergence lemma (Lemma C.3 page 45) enables us to pass to the limit in the right-hand side, since (w Dm ) m N converges weakly in L 2 ( (, T )) and (Π (1) u m ) m N converges strongly in L 2 ( (, T )). Owing to (5.55), this gives T T lim G Dm (x, t) (1) m u m (x, t)dxdt = T = and the proof of (5.54) is complete. w(x, t)u(x, t)dxdt G(x, t) u(x, t)dxdt We now state and prove the convergence of the fully implicit GS for (5.39). Theorem Assume (5.4) and let ((D T ) m ) m N be a sequence of space time GDs for homogeneous Dirichlet boundary conditions (with p = 2). Assume that the sequence ((D T ) m ) m N is space time-consistent, limit-conforming and compact in the sense of Definitions 4.3 and 4.6. We also suppose that ( Dm I Dm u ini ) m N is bounded in L 2 () d and, for any m N, we let u m be a solution to the GS (5.45). Then there exists a weak solution u of (5.39) in the sense of Definition 5.1 such that, up to a subsequence as m, sup t [,T ] Π (1) u m (t) u(t) L 2 (), and (1) u m u in L 2 ( (, T )) d. Proof. Let u, G and w be given by Lemma Then, sup t [,T ] Π (1) u m (t) u(t) L 2 () along a subsequence (not explicitly indicated below). Step 1: a strong monotonicity property. We aim to prove here that, for all V, W L 2 ( (, T )) d, T [µ( W )W µ( V )V ] [W V ] dxdt α W V 2 L 2 ( (,T )) d. (5.56)

164 5.2 Non-conservative problems 153 Use first the Cauchy Schwarz inequality for the dot product of R d to get T T µ( W )W V dxdt µ( W ) W V dxdt. Writing the same properties with W and V swapped leads to T (µ( W )W µ( V )V ) (W V ) dxdt T By Property (5.4e) on µ, (5.56) follows. Step 2: Proof that G = µ( u ) u. We use Minty s trick. For V L 2 ( (, T )) d, set T m (V ) = T (µ( W ) W µ( V ) V ) ( W V ) dxdt. [ ] [ ] µ( (1) u m ) (1) u m µ( V )V (1) u m V dxdt. Recall that µ( (1) u m ) (1) u m = G Dm. Together with (5.54), the weak convergences of G Dm and (1) u m therefore yield T lim T [ ] m(v ) = G µ( V )V [ u V ] dxdt. (5.57) m By (5.56), T m (V ) and thus T [ ] G µ( V )V [ u V ] dxdt. Take W L 2 ( (, T )) d and set V = u + λw for λ R. This gives T [ ] λ G µ( u + λw )( u + λw ) W dxdt. Since λ is any real number, this shows that the integral term is equal to zero. The dominated convergence theorem justifies letting λ in this term, which shows that T [ ] G µ( u ) u W dxdt =. Taking W = G µ( u ) u yields G = µ( u ) u a.e. on (, T ). (5.58) Step 3: strong convergence of (1) u m, and proof that u is a solution to (5.39).

165 154 5 Parabolic problems Making W = (1) u m and V = u in (5.56) gives (1) u m u 2 L 2 ( (,T )) d 1 α T m( u). By (5.57), lim m T m ( u) = and thus (1) u m u in L 2 ( (, T )) as m. This entails the convergences of the L 2 norms of these functions, that is (1) L u m u 2 ( (,T )) d L 2 ( (,T )) d as m. This latter convergence shows that the weak convergence of ( (1) u m ) m N to u in L 2 ( (, T )) is actually strong. By a form of non-linear strong convergence property similar to Lemma C.4 page 46, the strong convergences of (Π (1) u m ) m N and ( (1) u m ) m N show that, as m, ν(,, Π (1) u m, (1) u m ) ν(,, u, u) strongly in L 2 ( (, T )). The weak convergence of δ Dm u m towards t u and the weak-strong convergence property in Lemma C.3 page 45 then enable us to identify the limit of w Dm (defined by (5.51)): w = f ν(,, u, u) t u a.e. in (, T ). (5.59) Let v L 2 (, T ; H 1 ()) and take (v m ) m N provided by Lemma 4.9 for v. Write (5.53) for D T = (D T ) m and v = v m. Passing to the limit m in this relation is justified by the weak convergences of G Dm and w Dm, and the strong convergences of Π (1) v m and (1) v m. This leads to T G(x, t) v(x, t)dxdt = T w(x, t)v(x, t)dxdt. Then (5.58) and (5.59) show that u satisfies (5.41). Since the regularity properties of u required in Definition 5.1 are ascertained in Lemma 5.12, the proof that u is a solution to (5.39) is complete Semi-implicit scheme Given a space time gradient discretisation D T and using a semi-implicit timestepping, the GDM applied to (5.41) gives the following GS: seek a family u = (u (n) ) n=,...,n X N+1 D such that

166 5.2 Non-conservative problems 155 u () = I D u ini and, for n =,..., N 1, u (n+1) satisfies t (n+1) ν(x, t, Π D u (n), D u (n) )δ (n+ 1 2 ) D u(x)π D v(x)dxdt t (n) +δt (n+ 1 2 ) µ( D u (n) ) D u (n+1) (5.6) D v(x)dx = t (n+1) t (n) f(x, t)π D v(x)dxdt, v X D,. Quite naturally, the analysis of this semi-implicit implicit scheme uses both Π (1) D, (1) D and Π() D, () D. Recall that the definition of these latter operators (see (4.2)): Π () D u(x, t) = Π Du (n) (x) and () D u(x, t) = Du (n) (x), for a.e. (x, t) (t (n), t (n+1) ), n =,..., N 1. Estimates and existence of a solution to the semi-implicit scheme Lemma 5.15 (L 2 ( (, T )) estimate on δ D u and L (, T ; X D, ) estimate on u, semi-implicit scheme.). Under Hypotheses (5.4), let D T be a space time GD in the sense of Definition 4.1. Then the GS (5.6) has a unique solution u, and it satisfies, for all m = 1,..., N, t (m) ν min m 1 + µ min n= δ D u(x, t) 2 dxdt + µ min D u (m) 2 D u (n+1) (x) D u (n) (x) 2 dx L 2 () d µ max D I D u ini 2 L 2 () d + 1 ν min f 2 L 2 ( (,T )). Proof. First notice that, by Hypothesis (5.4e), (5.61) ξ, χ R d, χ ξ zµ(z)dz χ ξ 2 µ( ξ ) µ( ξ )χ (χ ξ). (5.62) To prove this property, simply remark by developing χ ξ 2 that it simplifies into ξ, χ R d 1 χ, 2 µ( ξ )( χ 2 ξ 2 ) zµ(z)dz. Set, for a, b R +, Φ(b) = 1 2 µ(a)(b2 a 2 ) b a ξ zµ(z)dz.

167 156 5 Parabolic problems Then Φ (b) = b(µ(a) µ(b)), whose sign is that of b a since µ is non-increasing. Hence Φ(b) is non-increasing for b a and non-decreasing for b a. Since Φ(a) =, this shows that Φ(b) for all b R + and the proof of (5.62) is complete. Applying this relation to ξ = D u (n) (x) and χ = D u (n+1) (x) and recalling the definition (5.42) of F leads to F ( D u (n+1) (x) ) F ( D u (n) (x) ) D u (n+1) (x) D u (n) (x) + µ min 2 µ( D u (n) (x) ) D u (n+1) (x) 2 [ D u (n+1) (x) D u (n) (x) ]. (5.63) Estimate (5.61) is then established as the proof of Lemma 5.11, by plugging v = u(n+1) u (n) in (5.6), summing over n =,..., m 1, and using (5.63) in δt (n+ 1 2 ) lieu of (5.48). Convergence of the semi-implicit scheme If u is the solution to the GS (5.6), let w D = f ν(π () D u D, () D u D)δ D u D, (5.64) G D = µ( () D u D ) (1) D u D, (5.65) Ĝ D = µ( () D u D ) () D u D. (5.66) Note that the GS (5.6) can be recast as: u X N+1 D, and, for all v X N+1 D,, T G D (x, t) (1) D v(x, t)dxdt T = w D (x, t)π (1) D v(x, t)dxdt. (5.67) The following lemma is the equivalent, for the semi-implicit scheme, of Lemma Lemma 5.16 (A convergence property of the semi-implicit scheme). Under Hypotheses (5.4), let ((D T ) m ) m N be a sequence of space time GDs for homogeneous Dirichlet boundary conditions (with p = 2). Assume that this sequence is space time-consistent, limit-conforming and compact in the sense of Definitions 4.3 and 4.6. Assume also that ( Dm I Dm u ini ) m N is bounded in L 2 () d. For m N, let u m be the solution to the GS (5.6), and define w Dm, G Dm and Ĝ from u m by, respectively, (5.64), (5.65) and (5.66).

168 5.2 Non-conservative problems 157 Then there exist functions u L (, T ; H 1 ()) C([, T ]; L 2 ()) with t u L 2 ( (, T )) and u(, ) = u ini, G L 2 ( (, T )) d, and w L 2 ( (, T )) such that, along a subsequence as m, sup t [,T ] Π (1) u m (t) u(t) L2 (), () u m and (1) u m converge weakly in L 2 ( (, T )) d to u, δ Dm u m converges weakly in L 2 ( (, T )) to t u, G Dm and Ĝ both converge weakly to G in L 2 ( (, T )) d, and T ( G Dm (x, t) Ĝ (x, t)) () D u m(x, t)dxdt, (5.68) w Dm converges weakly to w in L 2 ( (, T )), it holds T Ĝ Dm (x, t) () u m (x, t)dxdt T G(x, t) u(x, t)dxdt. (5.69) Proof. The proof is similar to the proof of Lemma The a priori estimate (5.61) provide the existence of u such that (Π (1) u m ) m N, ( (1) u m ) m N and (δ Dm u m ) m N converge as stated in the lemma. The same estimates show that ( G Dm ) m N and (Ĝ ) m N are bounded in L 2 ( (, T )) d, and therefore have weak limits in this space (up to a subsequence). Likewise, ( w Dm ) m N has a weak limit in L 2 ( (, T )) up to a subsequence. Let us now prove that () u m converges weakly to u, that the weak limits of ( G Dm ) m N and (Ĝ ) m N are the same, and that (5.68) holds. Since (1) u m () u m 2 = N 1 n= δt (n+ 1 2 ) N 1 δt Dm n= the estimate (5.61) shows that L 2 ( (,T )) d Dm u (n+1) m (x) Dm u (n) m (x) 2 dx Dm u m (n+1) (x) Dm u (n) m (x) 2 dx, (5.7)

169 158 5 Parabolic problems (1) u m () L2 u m as m. (5.71) ( (,T )) d This proves in particular that () u m u weakly in L 2 ( (, T )) d. Take now (ψ m ) m N bounded in L 2 ( (, T )) d and write T ( G Dm (x, t) Ĝ (x, t)) ψ m (x, t)dxdt T µ( () u m (x, t) ) (1) u m (x, t) () u m (x, t) ψ m (x, t) dxdt µ max (1) u m () u m L 2 ( (,T )) d ψ m L 2 ( (,T )) d. Use then (5.71) to infer T ( G Dm (x, t) Ĝ (x, t)) ψ m (x, t)dxdt as m. (5.72) Applied to ψ m = ψ for a fixed ψ, this relation that the weak limits of ( G Dm ) m N and (Ĝ ) m N are the same function G. The same relation (5.72) with ψ m = () u m provides (5.68). Let us conclude by proving (5.69). Relation (5.55) is established as in the proof of Lemma The GS (5.67) applied to D = and v = u m and the strong convergence of Π (1) u m then show that T T G Dm (x, t) (1) u m (x, t)dxdt = = T T w Dm (x, t)π (1) u m (x, t)dxdt w(x, t)u(x, t)dxdt G(x, t) u(x, t)dxdt. Since (1) u m () u m in L 2 ( (, T )) d and ( G Dm ) m N is bounded in L 2 ( (, T )) d, this gives T T G Dm (x, t) () u m (x, t)dxdt G(x, t) u(x, t)dxdt. We conclude the proof of (5.69) by using (5.68). The following theorem states the convergence of the semi-implicit scheme. The proof is omitted, as it is identical to the proof of Theorem 5.14, replacing G Dm by Ĝ and (1) u m by () D u m in the definition of T m (V ) in Step 2 (use of Minty trick).

170 5.3 Non-linear time-dependent Leray Lions problems 159 Theorem Assume (5.4) and let ((D T ) m ) m N be a sequence of space time GDs for homogeneous Dirichlet boundary conditions (with p = 2). Assume that ((D T ) m ) m N is space time-consistent, limit-conforming and compact in the sense of Definitions 4.3 and 4.6. We also suppose that ( Dm I Dm u ini ) m N is bounded in L 2 () d and, for any m N, we let u m be the solution to the GS (5.6). Then there exists a weak solution u of (5.39) in the sense of Definition 5.1 such that, up to a subsequence as m, sup t [,T ] Π (1) u m (t) u(t) L 2 (), and (1) u m u and () u m u in L 2 ( (, T )) d. 5.3 Non-linear time-dependent Leray Lions problems Model We consider here an evolution problem based on a Leray Lions operator, with non-homogeneous Neumann boundary conditions and non-local dependency on the lower order terms. The model reads t u div(a(x, u, u)) = f in (, T ), u(x, ) = u ini (x) in, a(x, u, u) n = g on (, T ), where a satisfies (3.96a) (3.96d) and T (, + ), u ini L 2 (), f L p ( (, T )) and g L p ( (, T )), where p = (5.73) (5.74) p p 1. The non-linear equation (5.73) covers a number of models, including semilinear ones appearing in image processing [18, 2]. The analysis of the GDM applied to (5.73) with homogeneous Dirichlet boundary conditions is done in [38]. The precise notion of solution to (5.73) is the following: u L p (, T ; W 1,p ()) C([, T ]; L 2 ()), u(, ) = u ini, t u L p (, T ; (W 1,p ()) ) and + T t u(, t), v(, t) (W 1,p ()),W 1,p ()dt a(x, u(, t), u(x, t)) v(x, t)dxdt T T = T f(x, t)v(x, t)dxdt + g(x, t)γv(x, t)ds(x)dt, v L p (; T ; W 1,p ()). (5.75)

171 16 5 Parabolic problems Remark The derivative t u is understood in the sense of distributions on (, T ) with values in L 2 (). Stating that it belongs to L p (, T ; (W 1,p ()) ) = (L p (, T ; W 1,p ())) amounts to asking that the linear form defined by C c (, T ; L 2 ()) L p (, T ; W 1,p ()) R ϕ t u, ϕ D (,T ;L 2 ()),D(,T ;L 2 ()) := = T T u(, t), t ϕ(, t) L 2 (),L 2 ()dt u(x, t) t ϕ(x, t)dxdt (5.76) is continuous for the norm of L p (, T ; W 1,p ()). Since the set of tensorial functions S = { q i=1 ϕ i(t)β i (x) : q N, ϕ i Cc (, T ), β i C ()} is dense in L p (, T ; W 1,p ()) (see [31, Corollary 1.3.1]), the derivative t u belongs to L p (, T ; (W 1,p ()) ) if and only if (5.76) is continuous on S for the L p (, T ; W 1,p ())-norm. Remark Using regularisation and integration-by-parts techniques [31, Section 2.5.2], it is possible to see that any solution u to (5.75) also satisfies, for any s [, T ], 1 s 2 u(s) 2 L 2 () + a(x, u(, τ), u(x, τ))dxdτ = 1 s 2 u ini 2 L 2 () + f(x, τ)u(x, τ)dxdτ s + g(x, τ)γu(x, τ)ds(x)dτ. (5.77) With a reasoning similar to the one employed to establish the equivalence of (5.3) and (5.4), we can see that (5.75) is equivalent to: u L p (, T ; W 1,p ()) L (, T ; L 2 ()) and, for all v C 1 ([, T ]; W 1,p () L 2 ()) such that v(, T ) =, T u(x, t) t v(x, t)dxdt u ini (x)v(x, )dx T (5.78) + a(x, u(, t), u(x, t)) v(x, t)dxdt T = T f(x, t)v(x, t)dxdt + g(x, t)γv(x, t)ds(x)dt. To prove this equivalence, we use [31, Section 2.5.2] to see that if u L p (, T ; W 1,p () L 2 ()) satisfies t u L p (, T ; (W 1,p () L 2 ()) ), then u C([, T ]; L 2 ()). We also use the density in L p (, T ; W 1,p ()) of C 1 ([, T ]; W 1,p () L 2 ()), which is for example a consequence of [31, Corollary 1.3.1].

172 5.3 Non-linear time-dependent Leray Lions problems Gradient scheme and main results Let D T = (X D, Π D, T D, D, I D, (t (n) ) n=,...,n ) be a space time GD for nonhomogeneous Neumann conditions in the sense of Definition 4.1, and let θ [ 1 2, 1]. The GDM applied to Problem (5.73) yields the following GS: find a family (u (n) ) n=,...,n X N+1 D such that u () = I D u ini X D and, for all n =,..., N 1, u (n+1) satisfies δ (n+ 1 2 ) D u(x)π D v(x)dx ( ) + a x, Π D u (n+θ), D u (n+θ) (x) D v(x)dx 1 t (n+1) (5.79) = f(x, t)π δt (n+ 1 2 ) D v(x)dxdt t (n) + 1 t (n+1) g(x, t)t δt (n+ 1 2 ) D v(x)ds(x)dt, v X D. t (n) The choice θ 1 2 is required for stability reasons. As explained in Section 4.1, θ = 1 leads to the classical Euler time implicit discretisation, while θ = 1 2 corresponds to the Crank-Nicholson time discretisation. Recalling the notations in (4.2), we now state our initial convergence results for this GS. Theorem 5.2 (Convergence of the GS for transient Leray Lions). Under Assumptions (3.96a) (3.96d) and (5.74), let ((D T ) m ) m N be a sequence of space time GDs for non-homogeneous Neumann boundary conditions, in the sense of Definition 4.1. Assume that this sequence is space timeconsistent, limit-conforming and compact in the sense of Definitions 4.3 and 4.6. Let θ [ 1 2, 1] be given. Then, for any m N, there exists a solution u m to the GS (5.79) with D = and, along a subsequence as m, Π (θ) u m converges to u strongly in L p ( (, T )), Π (1) u m converges to u weakly in L 2 () uniformly on [, T ] (see Definition 4.29), (θ) u m converges to u weakly in L p ( (, T )) d, where u is a solution to (5.75). Remark As for the stationary problem (see Remark 3.35), the existence of a solution to (5.75) is a by-product of the proof of convergence of the GDM. Moreover, in the case where the solution u of (5.75) is unique, the whole sequence (u m ) m N converges to u in the senses above. The convergence of the function reconstructions is actually much better than in the initial result above. It is uniform-in-time and strong in space.

173 162 5 Parabolic problems Theorem 5.22 (Uniform-in-time convergence of the GS). Under the assumptions and notations of Theorem 5.2, and along the same subsequence as in this theorem, we have sup t [,T ] Π (θ) u m (t) u(t) L 2 (), sup t [,T ] Π (1) u m (t) u(t) L 2 (). If the Leray Lions operator a is strictly monotone, then a strong convergence result can also be stated on the gradients. Theorem 5.23 (Strong convergence of the gradients in the strictly monotone case). Let us assume the hypotheses of Theorem 5.2, and that a is strictly monotone in the sense of (3.98). Then, with the same notations and along the same subsequence as in Theorem 5.2, (θ) u m converges strongly to u in L p ( (, T )) d A priori estimates We begin by establishing a priori estimates. Lemma 5.24 (L (, T ; L 2 ()) estimate, discrete L p (, T ; W 1,p ()) estimate, and existence of a solution to the GS). Under Hypotheses (3.96a) (3.96d) and (5.74), let D T be a space time GD for non-homogeneous Neumann conditions in the sense of Definition 4.1. Then there exists at least one solution to the GS (5.79), and there exists C 12 >, depending only on p, C P C D, C ini Π D I D u ini L2 (), f, g and a such that, for any solution u to this scheme, sup Π (1) t [,T ] and (θ) D u(t) L 2 () C 12, sup D u L p ( (,T )) d C 12. Π (θ) t [,T ] D u(t) L 2 () C 12 Proof. Let us first prove the estimates. Recall (5.22), that is δt (n+ 1 2 ) δ (n+ 1 2 ) D u Π D u (n+θ) 1 ( (Π D u (n+1) ) 2 (Π D u (n) ) 2), 2 (5.8) choose v = δt (n+ 1 2 ) u (n+θ) in (5.79), and sum on n =,..., k 1 for a given k {1,..., N}. This yields 1 Π D u (k) L 2 () 1 Π D u () 2 2 t (k) L 2 () ( a t (k) + x, Π (θ) D ) u(, t), (θ) D u(x, t) f(x, t)π (θ) D u(x, t)dxdt (θ) D u(x, t)dxdt

174 + t (k) 5.3 Non-linear time-dependent Leray Lions problems 163 g(x, t)t (θ) D u(x, t)ds(x)dt. (5.81) In particular, owing to the coercivity property (3.96b) of a, and using Hölder s inequality and Young s inequality (C.9) (the latter with ε = Π D u (k) 2 L 2 () Π D u () 2 + a 4C p D + a 4C p D Π (θ) D L 2 () t (k) + a (θ) D u(, t) p dt L p () d + 41/(p 1) C p D (pa) 1/(p 1) p f p u p T (θ) D u p L p ( (,t (k) )) L p ( (,t (k) )) L p ( (,t (k) )) + 41/(p 1) C p D (pa) 1/(p 1) p g p. pa 4C p D ), L p ( (,t (k) )) Apply the definition (2.26) of C D and recall that u () = I D u ini to deduce 1 Π D u (k) 2 + a (θ) 2 L 2 D () 2 u p L p ( (,t (k) )) d 41/(p 1) C p D 1 2 Π DI D u ini 2 L 2 () + (pa) 1/(p 1) p f p + 41/(p 1) C p D (pa) 1/(p 1) p g p L p ( (,t (k) )). L p ( (,t (k) )) This establishes the estimates on Π (1) D u and (θ) D u. The estimate on Π(θ) D u follows from the inequality Π D u (n+θ) L2 θ Π D u (n+1) L2 + (1 θ) Π D u (n) L2. () () () The existence of at least one solution to (5.79) is done as in the proof of Theorem 3.34, reasoning on u (n+θ) and using the above estimates. The following estimate will be useful to apply the Aubin Simon theorem for GD (Theorem 4.21). Lemma 5.25 (Estimate on the dual norm of the discrete time derivative). Under Hypotheses (3.96a) (3.96d) and (5.74), let D T be a space time GD for non-homogeneous Neumann conditions in the sense of Definition 4.1. Let u be a solution to the GS (5.79). Then there exists C 13, depending only on p, µ, a, a, C ini Π D I D u ini L 2 (), f, g, T and C P C D, such that T δ D u(t) p,d dt C 13, (5.82) where the dual norm,d is given by Definition 4.18.

175 164 5 Parabolic problems Proof. Let us take a generic v X D as a test function in (5.79). We have, thanks to Assumption (3.96d) on a, δ (n+ 1 2 ) D u(x)π D v(x)dx ( a(x) + µ D u (n+θ) (x) p 1) D v(x) dx 1 + δt (n+ 1 2 ) t (n+1) t (n) 1 + δt (n+ 1 2 ) f(x, t)π D v(x)dxdt t (n+1) t (n) g(x, t)t D v(x)ds(x)dt. This leads, by definition (2.26) of C D, to the existence of C 14 > depending only on p, µ such that δ (n+ 1 2 ) D u(x)π D v(x)dx a L p () + D u (n+θ) p 1 L p () d C 14 + C t (n+1) D f(, t) δt (n+ 1 2 ) L p () dt v D. t (n) + C t (n+1) D g(, t) δt (n+ 1 2 ) L p ( ) dt t (n) Taking the supremum on v X D such that v D = 1 gives δ (n+ 1 2 ) u C 14 a L p () + C 14 D u (n+θ) p 1,D D L p () d + C t (n+1) 14C D f(, t) δt (n+ 1 2 ) L p () dt + C t (n+1) 14C D g(, t) t (n) δt (n+ 1 2 ) L p ( ) dt. t (n) The proof is concluded by raising this estimate to the power p, distributing this power to each term on the right-hand side thanks to the powerof-sums inequality (C.14), using Jensen s inequality for the integral terms, multiplying by δt (n+ 1 2 ), summing on n and invoking Lemma 5.24 to estimate (θ) D u p L p ( (,T )) d Proof of the convergence results We now prove the convergence of the GDM for the transient Leray Lions model (5.73). Proof of Theorem 5.2.

176 5.3 Non-linear time-dependent Leray Lions problems 165 Step 1: Application of compactness results. The definition (2.25) of Dm and Estimates (5.8) and (5.82) show that the hypotheses of Lemma 4.7 (regularity of the limit) and Theorem 4.21 (Aubin Simon for GD) are satisfied by (u m ) m N. There exists therefore u L p (, T ; W 1,p ()) such that, up to a subsequence as m, (Π (θ) u m u strongly in L p ( (, T )), (θ) u m u weakly in L p ( (, T )) d, and T (θ) u m γu weakly in L p ( (, T )). Moreover, since (Π (θ) u m ) m N is bounded in L (, T ; L 2 ()), the convergence Π (θ) u m u also holds in L (, T ; L 2 ()) weak-. Estimates (5.8) and (5.82) show that (u m ) m N satisfies the assumptions of Theorem 4.32 with θ = 1. Hence, up to a subsequence as m, (Π (1) u m ) m N converges to some ũ uniformly-in-time for the weak topology of L 2 () (as per Definition 4.29). Estimates (5.8) and Assumption (3.96d) show that the functions A Dm (x, t) = a(x, Π (θ) u m (, t), (θ) u m (x, t)) remain bounded in L p ( (, T )) d. Up to a subsequence, A Dm therefore converges to some A weakly in L p ( (, T )) d as m. Step 2: Proof that u = ũ. First notice that the convergence of (Π (1) u m ) m N towards ũ also holds for the weak topology of L 2 ( (, T )) (this is an easy consequence of its convergence uniformly-in-time and weakly in L 2 ()). Take ϕ Cc ( (, T )) and let ( P Dm ϕ(t) = argmin Π Dm w ϕ(t) L max(p,2) () w X Dm Since X Dm, a triangle inequality shows that + Dm w ϕ(t) L p () d ). Π Dm P Dm ϕ(t) L max(p,2) () + P Dm ϕ(t) Lp () d 2 ϕ(t) L max(p,2) () + 2 ϕ(t) L p () d. (5.83) In particular, by definition of Dm and smoothness of ϕ, P Dm ϕ Lp (,T ;X Dm ) remains bounded. Moreover, the space time-consistency of ((D T ) m ) m N ensures that, for all t (, T ), Π Dm P Dm ϕ(t) ϕ(t) in L 2 () as m. Combined with the dominated convergence theorem and (5.83), this yields Π Dm P Dm ϕ ϕ in L 2 ( (, T )) as m. For n {,..., N 1} and t (t (n), t (n+1) ], Π (1) u m (t) Π (θ) u m (t) = Π Dm u m (n+1) Π Dm u (n+θ) m = (1 θ)(π Dm u (n+1) m Π Dm u (n) m )

177 166 5 Parabolic problems and thus, by (4.29), T = (1 θ)δt (n+ 1 2 ) δ Dm u m (t) (5.84) ( ) Π (1) u m (x, t) Π (θ) u m (x, t) Π Dm [P Dm ϕ(t)](x)dxdt T (1 θ)δt Dm δ Dm u(t),dm P Dm ϕ(t) Dm dt. Use Lemma 5.25 and Hölder s inequality to see that the right-hand side of this relation tends to as m. Since Π (1) u m and Π (θ) u m converge weakly in L 2 ( (, T )) towards ũ and u, respectively, we deduce T = lim m (ũ(x, t) u(x, t))ϕ(x, t)dxdt T (Π (1) u m (x, t) Π (θ) u m (x, t))π Dm [P Dm ϕ(t)](x)dxdt =. (5.85) This proves that ũ = u, and thus that Π (1) u m u uniformly on [, T ] weakly in L 2 (). In particular, Π (1) u m (T ) u(t ) weakly in L 2 () and thus u(x, T ) 2 dx lim inf m Π (1) u m (x, T ) 2 dx. (5.86) Step 3: Proof that u is a solution to (5.75). Let v C 1 ([, T ]; W 1,p () L 2 ()) such that v(, T ) =, and let (v m ) m N be given by Lemma 4.9. Properties (4.7), (4.8) and (4.1) therefore hold, with θ =. We drop some indices m for legibility. Using δt (n+ 1 2 ) v (n) as test function in (5.79) yields T (m) 1 + T (m) 2 = T (m) 3 + T (m) 4 with and T (m) 1 = N 1 n= [ ΠD u (n+1) (x) Π D u (n) (x) ] Π D v (n) (x)dx, T ( ) T (m) 2 = a x, Π (θ) D u(, t), (θ) D u(x, t) T T (m) 3 = T T (m) 4 = f(x, t)π () D v(x, t)dxdt, g(x, t)t () D v(x, t)ds(x)dt. () D v(x, t)dxdt,

178 5.3 Non-linear time-dependent Leray Lions problems 167 Accounting for v (N) =, the discrete integrate-by-parts formula (C.15) gives T (m) 1 = = N 1 n= T Π D u (n+1) (x) [ Π D v (n+1) (x) Π D v (n) (x) ] dx Π D u () (x)π D v () (x)dx Π (1) D u(x, t)δ Dv(x, t)dxdt Π D I D u ini (x)π () D v(x, )dx. The strong convergences (4.1c) and (4.1b) of v m and the weak convergence in L 2 ( (, T )) of Π (1) u m thus ensure that, as m, T T (m) 1 u(x, t) t v(x, t)dxdt u ini (x)v(x, )dx. (5.87) Owing to the weak convergence of A Dm and the strong convergence (4.7b) of () v m, as m, T T (m) 2 A(x, t) v(x, t)dxdt. (5.88) Finally, by (4.7a) and (4.8), as m, T T (m) 3 T (m) 4 T f(x, t)v(x, t)dxdt, g(x, t)γv(x, t)ds(x)dt. and (5.89) Using (5.87) (5.89) we can pass to the limit in T (m) 1 + T (m) 2 = T (m) 3 + T (m) 4 to see that T u(x, t) t v(x, t)dxdt u ini (x)v(x, )dx T + A(x, t) v(x, t)dxdt T = T f(x, t)v(x, t)dxdt + g(x, t)γv(x, t)ds(x)dt. This holds for all v C 1 ([, T ]; W 1,p () L 2 ()) such that v(, T ) =. By a density argument similar to the one used to prove the equivalence of (5.3) and (5.4), or of (5.75) and (5.78), we infer that u L p (, T ; W 1,p ()) C([, T ]; L 2 ()), t u L p (, T ; (W 1,p ()) ), u(, ) = u ini and, for all v L p (, T ; W 1,p ()),

179 168 5 Parabolic problems T T t u(, t), v(, t) (W 1,p ()),W 1,p ()dt + = T It remains to prove that A(x, t) v(x, t)dxdt T f(x, t)v(x, t)dxdt + g(x, t)γv(x, t)ds(x)dt. (5.9) A(x, t) = a(x, u(, t), u(x, t)), for a.e. (x, t) (, T ). (5.91) The formula T t u(, t), u(, t) (W 1,p ()),W 1,p ()dt = 1 u(x, T ) 2 dx u(x, ) 2 dx is justified by [31, Section 2.5.2] since u L p (, T ; W 1,p () L 2 ()) and t u L p (, T ; (W 1,p ()) ). Writing (5.9) with v = u thus yields 1 u(x, T ) 2 dx T = f(x, t)u(x, t)dxdt + T u ini (x) 2 dx + T A(x, t) u(x, t)dxdt g(x, t)γu(x, t)ds(x)dt. (5.92) Relation (5.81) with k = N yields 1 (Π (1) D 2 u(x, T ))2 dxdt + T T ( a x, Π (θ) D u(, t), (θ) T (Π D I D u ini (x)) 2 dx + D u(x, t) ) g(x, t)t (θ) D u(x, t)ds(x)dt. (θ) D u(x, t)dxdt f(x, t)π (θ) D u(x, t)dxdt (5.93) Recall that Π (θ) u m u strongly in L p ( (, T )), that T (θ) u m γu weakly in L p ( (, T )) and, by space time consistency, that Π Dm I Dm u ini u ini in L 2 (). Moving the first term of (5.93) into the right-hand side, taking the superior limit of the resulting inequality and using (5.86) therefore leads to T ( lim sup a m lim inf m 1 2 x, Π (θ) D ) u(, t), (θ) D u(x, t) (Π (1) D u(x, T ))2 dxdt (θ) D u(x, t)dxdt u ini (x) 2 dx

180 T T 5.3 Non-linear time-dependent Leray Lions problems 169 T f(x, t)u(x, t)dxdt + g(x, t)γu(x, t)ds(x)dt u ini (x) 2 dx u(x, T ) 2 dxdt f(x, t)u(x, t)dxdt + Relation (5.92) then yields T T ( lim sup a x, Π (θ) u m (, t), (θ) u m (x, t) m T ) g(x, t)γu(x, t)ds(x)dt. (θ) u m (x, t)dxdt A(x, t) u(x, t)dxdt. (5.94) It is now possible to apply Minty s trick. Consider, for G L p (, T ; L p ()) d, the quantity T [ ( ) a x, Π (θ) D u(, t), (θ) D u(x, t) a ( x, Π (θ) D u(, t), G(x, t) )] [ (θ) D u(x, t) G(x, t) ] dxdt. (5.95) Since Π (θ) u m u strongly in L p (, T ; L p ()), up to a subsequence we can assume that Π (θ) u m (t) u(t) strongly in L p () for a.e. t (, T ). Assumptions (3.96a) and (3.96d) and the dominated convergence theorem then show that a(, Π (θ) D u, G) a(, u, G) strongly in Lp ( (, T )) d. Developping (5.95), all the terms except one pass to the limit by weak-strong convergence (cf. Lemma C.3). For the only weak-weak limit, apply (5.94) and, taking the superior limit as m, write T [A(x, t) a (x, u(, t), G(x, t))] [ u(x, t) G(x, t)]dxdt. In a similar way as in Step 2 of the proof of Theorem 3.34, take then G = u + αϕ for α R and ϕ L p (, T ; L p ()) d, divide by α and let α. This gives T [A(x, t) a (x, u(, t), u(x, t))] ϕ(x, t)dx =, which shows that (5.91) holds. The proof that u is a weak solution to (5.75) is therefore complete. Proof of Theorem Step 1: a preliminary result.

181 17 5 Parabolic problems Take (s m ) m N [, T ] that converges to some s [, T ]. Since Π (θ) u m u strongly in L p ( (, T )), as in Step 3 of the proof of Theorem 5.2, Assumptions (3.96a) and (3.96d) show that (1 [,sm]a(x, Π (θ) u m, u)) m N converges strongly in L p ( (, T )) d. The weak convergence of ( (θ) u m ) m N to u in L p ( (, T )) d then yields sm ( ) a x, Π (θ) u m (, t), u(x, t) [ (θ) u m (x, t) u(x, t) ] dxdt. (5.96) Write (5.95) with s m instead of T and u instead of G, and develop the terms. Using (5.96), the weak convergence a(, Π (θ) u m, (θ) u m ) a(, u, u) in L p ( (, T )) (see (5.91)), and the strong convergence 1 [,sm] u 1 [,s] u in L p ( (, T )), we obtain sm ( lim inf a x, Π (θ) m u m (, t), (θ) u m (x, t) s ) (θ) u m (x, t)dxdt a (x, u(, t), u(x, t)) u(x, t)dxdt. (5.97) Step 2: proof of the uniform-in-time strong in L 2 () convergences. Let s [, T ] and k(s) such that s (t (k(s)), t (k(s)+1) ]. Apply (5.81) to k = k(s) + 1 to write 1 Π (1) D 2 m u m (s) 2 L 2 () s + a(x, Π (θ) u m (, t), (θ) u m (x, t)) (θ) u m (x, t)dxdt 1 s 2 Π I Dm u ini 2 L 2 () + f(x, t)π Dm u m (x, t)dxdt s + g(x, t)t Dm u m (x, t)ds(x)dt + ρ(δt Dm ) (5.98) where ρ(δt Dm ) as δt Dm (all time integrals should be up to t (k(s)+1), but we used the non-negativity of the integrand involving a to limit its integral to s, and ρ is the quantity that includes the remaining parts of the integrals in the right-hand side, estimated using to (5.8)). The proof of the uniform convergence of (Π (1) u m ) m N is done by invoking Lemma As in Step 1, take (s m ) m N [, T ] that converges to some s [, T ]. We want to show that Π (1) u m (s m ) u(s) in L 2 (). Apply (5.98) with s = s m, move the second term to the right-hand side, and take the superior limit as m. Relation (5.97) and the strong (resp. weak) convergence of

182 5.3 Non-linear time-dependent Leray Lions problems 171 Π (θ) u m (resp. T (θ) u m ) enable us to pass to the limit in all the terms except the first one. Owing to (5.77), this gives 1 lim sup Π (1) D m 2 m u m (s m ) 2 L 2 () s a(x, u(, t), u(x, t)) u(x, t)dxdt + 1 s 2 u ini 2 L 2 () + f(x, t)u(x, t)dxdt s + g(x, t)γu(x, t)ds(x)dt = 1 2 u(s) 2 L 2 (). (5.99) The uniform-in-time weak L 2 () convergence of (Π (1) u m ) m N towards u and Lemma 4.28 show that Π (1) u m (s m ) u(s) in L 2 () weak. Owing to (5.99), this convergence is actually strong in L 2 (). Invoke again Lemma 4.28 to conclude that sup t [,T ] Π (1) u m (t) u(t) L 2 (). The strong convergence of (Π (θ) u m ) m N in the same sense follows immediately from the definition of these functions, the strong convergence of (Π (1) u m ) m N, and the continuity of u : [, T ] L 2 (). Indeed, Π (θ) u m (, t) is a convex combination of values of Π (1) u m at two times within distance δt Dm of t. Proof of Theorem Using (5.94) and (5.96) with s m = T, T [ ( lim sup a x, Π (θ) u m, (θ) m u m ) a [ (θ) u m u ] dxdt. ( ) ] x, Π (θ) u m, u This relation and the strict monotonicity of a enable us to conclude, as in Step 3 of the proof of Theorem 3.34, that (θ) u m u a.e. on (, T ). From (5.94) and (5.91) we also infer T ( ) lim sup a x, Π (θ) u m, (θ) u m (θ) u m dxdt m T a (x, u, u) udxdt. Together with (5.97) (with s m = T ), this proves that this relation holds with a limit instead of a superior limit, and an equality instead of an inequality. The same technique as in Step 3 of the proof of Theorem 3.34 then yields the strong convergence of Dm u m to u in L p ( (, T )) d.

183

184 6 Degenerate parabolic problems In this chapter, we study the following generic nonlinear parabolic model t β(u) div (Λ(x) ζ(u)) = f in (, T ), β(u)(x, ) = β(u ini )(x) in, ζ(u) = on (, T ), (6.1) where β and ζ are non-decreasing. This model arises in various frameworks (see next section for precise hypotheses on the data). This model includes 1. Richards model, setting ζ(s) = s, which describes the flow of water in a heterogeneous anisotropic underground medium, 2. Stefan s model [8], setting β(s) = s, which arises in the study of a simplified heat diffusion in a melting medium. The purpose of this chapter is to study the convergence of gradient schemes for (6.1). Although Richards and Stefan s models are formally equivalent when β and ζ are strictly increasing (consider β = ζ 1 to pass from one model to the other), they change nature when these functions are allowed to have plateaux. Stefan s model can degenerate to an ODE (if ζ is constant on the range of the solution), and Richards model can become a non-transient elliptic equation (if β is constant on this range). Remark 6.1. The techniques developed in this chapter also apply to the following more general non-linear PDE, which mixes (6.1) and Leray Lions operators as in Section 5.3: t β(u) diva (x, ν(u), ζ(u)) = f, (6.2) where ν = β ζ. We refer to [35] for the analysis of gradient schemes for (6.2). The chapter is organised as follows. In Section 6.1, we present the assumptions and the notion of weak solution for (6.1), and we show that this problem

185 174 6 Degenerate parabolic problems can be reformulated using the notion of maximal monotone graph. Section 6.2 presents the gradient schemes (GSs) obtained by applying the gradient discretisation method (GDM) to the generic model (6.1). Based on estimates proved in Section 6.3, Section 6.4 contains the convergence proof of these GSs. Section 6.5 is focused on a uniform-in-time convergence result. A uniqueness result, based on the existence of a solution to the adjoint problem, is given in Section 6.7. Numerical examples, presented in Section 6.8 and based on the VAG scheme (see Section 8.5), complete this chapter. 6.1 The continuous problem Hypotheses and notion of solution We consider the evolution problem (6.1) under the following hypotheses. is an open bounded connected polytopal subset of R d (d N ) and T >, (6.3a) ζ : R R is non-decreasing, Lipschitz continuous with Lipschitz constant L ζ >, ζ() = and, for some M, M 1 >, ζ(s) M s M 1 for all s R, (6.3b) β : R R is non-decreasing, Lipschitz continuous with Lipschitz constantl β >, and β() =, (6.3c) β + ζ is strictly increasing, (6.3d) Λ : M d (R) is measurable and there exists λ λ > such that, for a.e. x, Λ(x) is symmetric with eigenvalues in [λ, λ]. (6.3e) u ini L 2 (), f L 2 ( (, T )). (6.3f) Remark 6.2 (Common plateaux of ζ and β) Hypothesis (6.3d) does not restrict the generality of the model. Indeed, if we only assume (6.3b)-(6.3c), and if there exist s 1 < s 2 such that (β + ζ)(s 1) = (β + ζ)(s 2), then [s 1, s 2] is a common plateau of β and ζ. Denoting by β, ζ and ν the functions obtained from β and ζ by removing this common plateau (by a contraction of the s-ordinate), we see that u is a solution to (6.1) if and only if u is a solution of the same problem with β and ζ replaced with β and ζ. The precise notion of solution to (6.1) that we consider is the following:

186 6.1 The continuous problem 175 ζ(u) L 2 (, T ; H 1 ()), T β(u)(x, t) t v(x, t)dxdt β(u ini (x))v(x, )dx T + Λ(x) ζ(u)(x, t) v(x, t)dxdt (6.4) T = f(x, t)v(x, t)dxdt, v L 2 (; T ; H 1 ()) such that t v L 2 ((, T ) ) and v(, T ) =. Remark 6.3 (All the terms in (6.4) make sense). If v and t v belong to L 2 (, T ; L 2 ()), then v C([, T ]; L 2 ()) (see [31]), and we can therefore impose the pointwise-in-time value of v(, T ). Moreover, Assumptions (6.3b) and (6.3c) ensure that, if ζ(u) L 2 ((, T ) ), then u and β(u) also belong to L 2 ((, T ) ). Hence, all the terms in (6.4) are well-defined. The existence of a solution to this problem follows from the proof of convergence of the GS (see Remark 6.15). The uniqueness of this solution is proved in Section 6.7. Theorem 6.4 (Existence and uniqueness of the weak solution). Under Hypotheses (6.3), there exists a unique solution to (6.4). Remark 6.5. We will see in Corollary 6.17 that the solution to (6.4) enjoys additional regularity properties, and that (6.4) can be recast in a stronger form A maximal monotone operator viewpoint Following [4], we show here that (6.1) can be recast in a maximal monotone operator framework. Lemma 6.6 (Maximal monotone operator). Let T : R R be a multivalued operator, that is a function from R to the set P(R) of all subsets of R. The following properties are equivalent: 1. T is a maximal monotone operator with domain R, T () and T is sublinear in the sense that there exist T 1, T 2 such that, for all x R and all y T (x), y T 1 x + T 2 ; 2. There exist ζ and β satisfying (6.3b) and (6.3c) such that the graph of T is given by Gr(T ) = {(ζ(s), β(s)), s R}. Proof. (2) (1). Clearly = (ζ(), β()) T (). The monotonicity of T follows from the fact that ζ and β are nondecreasing. We now have to prove that T is maximal, that is, if x, y satisfy (ζ(s) x)(β(s) y) for all s R then (x, y) Gr(T ). By (6.3b) and (6.3c), the mapping β +ζ : R R is onto, so there exists s R such that

187 176 6 Degenerate parabolic problems β(s) + ζ(s) = x + y. (6.5) Then ζ(s) x = y β(s) and therefore (β(s) y) 2 = (ζ(s) x)(β(s) y). This implies β(s) = y and, combined with (6.5), ζ(s) = x. Hence (x, y) Gr(T ). The sub-linearity of T follows from β(s) L β s L β ( ζ(s) + M 2 )/M 1. (1) (2). Recall that the resolvent R(T ) = (Id + T ) 1 of the maximal monotone operator T is a single-valued function R R that is nondecreasing and Lipschitz continuous with Lipschitz constant 1. Set ζ = R(T ) and β = Id ζ. These functions are nondecreasing and Lipschitz continuous with constant 1. By definition of the resolvent, (x, y) Gr(T ) (x, x+y) Gr(Id+T ) (x+y, x) Gr(ζ) x = ζ(x+y). Since β = Id ζ, setting s = x + y shows that (x, y) Gr(T ) is equivalent to (x, y) = (ζ(s), β(s)). Since T () this gives β() = ζ() =. Finally, the existence of M 1 and M 2 in (6.3b) follows from the sublinearity of T. If (x, y) Gr(T ) then y T 1 x + T 2 and x = ζ(x + y), which gives x + y ((1 + T 1 ) ζ(x + y) + T 2 ). Using this lemma, we recast (6.1) as t T (z) div (Λ(x) z) = f in (, T ), T (z)(, ) = b ini in, z = on (, T ) (6.6) where b ini = β(u ini ) L 2 (). Hypotheses (6.3c) and (6.3b) are translated into: T : R P(R) is a maximal monotone operator, T () and T is sublinear: T 1, T 2 such that, for all x R all y T (x), y T 1 x + T 2. (6.7) Definition 6.7. Let us assume (6.3a), (6.3e), (6.3f) and (6.7). Let z ini L 2 () and b ini : R such that, for a.e. x, b ini (x) T (z ini (x)). A solution to (6.6) is a pair of functions (z, b) satisfying z L 2 (, T ; H 1 ()), b(x, t) T (z(x, t)) for a.e. (x, t) (, T ), T b(x, t) t v(x, t)dxdt b ini (x)v(x, )dx T + Λ(x) z(x, t) v(x, t)dxdt (6.8) T = f(x, t)v(x, t)dxdt v L 2 (; T ; H 1 ()) such that t v L 2 ((, T ) ) and v(, T ) =.

188 6.3 Estimates on the approximate solution 177 Remark 6.8. The sublinearity of T ensures that b L 2 (, T ; L 2 ()) and b ini L 2 (), since z L 2 (, T ; L 2 ()) and z ini L 2 (). Given (z ini, b ini ) as in Definition 6.7 and fixing ζ = R(T ) and β = Id ζ (as in the proof of Item 2 of Lemma 6.6), we can find a measurable u ini such that z ini = ζ(u ini ) and b ini = β(u ini ). the estimate z ini M u ini M 1 ensures that u ini L 2 (). These ζ, β and u ini being fixed, the existence and uniqueness of the solution to (6.4) (Theorem 6.4) gives the existence and uniqueness of the solution to (6.8). This solution satisfies that z = ζ(u) and b = β(u), where u is the unique solution to (6.4). 6.2 Gradient scheme Let p = 2 and D T = (D, I D, (t (n) ) n=,...,n ) be a space time gradient discretisation for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Assume that D has the piecewise constant reconstruction property in the sense of Definition 2.1. We take θ = 1 in (4.2), which means that an implicit time-stepping is considered. We recall the corresponding notations Π (1) D and (1) D. Formally integrating (6.4) by parts in time, we obtain a new formulation of (6.1) (see (6.25)). The GDM applied to (6.4) leads to a GS which merely consists in using, in this new formulation, the discrete space and mappings of the GD. The GS is therefore: seek a family (u (n) ) n=,...,n X D, such that u () = I D u ini and, for all v = (v (n) ) n=1,...,n X D,, T [ δ D β(u)(x, t)π (1) D v(x, t) + Λ(x) (1) D ζ(u)(x, t) (1) D ]dxdt v(x, t) T = f(x, t)π (1) D v(x, t)dxdt. (6.9) We recall the definition, in Remark 2.11, of ζ(u) and β(u), which is coherent with Π D since this reconstruction is piecewise constant. Remark 6.9 (Crank Nicolson and θ-scheme) As in Section 5.3, we could as well consider, instead of a fully implicit time-stepping, a Crank Nicolson scheme or any scheme in between those two. Such a scheme is defined by taking θ [ 1, 1] in (4.2). All the results we establish for (6.9) would hold 2 for such a scheme. 6.3 Estimates on the approximate solution As it is usual in the study of numerical methods for PDE with strong nonlinearities or without regularity assumptions on the data, everything starts with a priori estimates.

189 178 6 Degenerate parabolic problems Lemma 6.1 (L (, T ; L 2 ()) estimate and discrete L 2 (, T ; H 1 ()) estimate). Under Assumptions (6.3), let D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Assume that the underlying spatial discretisation has a piecewise constant reconstruction in the sense of Definition 2.1, and that u is a solution to the corresponding GS (6.9). Let η : R R be defined by s R, η(s) = s ζ(q)β (q)dq. (6.1) Let T (, T ] and denote by k = 1,..., N the index such that T (t (k 1), t (k) ]. Then T Π (1) D η(u)(x, T )dx + Π D η(i D u ini )(x)dx + Λ(x) (1) D t (k) ζ(u)(x, t) (1) D ζ(u)(x, t)dxdt f(x, t)π (1) D ζ(u)(x, t)dxdt. (6.11) Consequently, there exists C 1 >, depending only on L β, L ζ, C P C D (see Definition 2.2), C ini Π D I D u ini L2 (), f and λ such that sup Π (1) t [,T ] and D η(u)(t) L C 1, 1 () sup Π (1) t [,T ] D β(u)(t) L 2 () C 1. (1) D ζ(u) L 2 ( (,T )) d C 1 (6.12) Proof. Let us first remark that, for all a, b R, an integration by parts gives η(b) η(a) = b a ζ(q)β (q)dq = ζ(b)(β(b) β(a)) b a ζ (q)(β(q) β(a))dq. Since b a ζ (q)(β(q) β(a))dq (as ζ and β are non-decreasing), we get η(b) η(a) ζ(b)(β(b) β(a)). (6.13) Using Remark 2.11 (consequence of the definition 2.1 of piecewise constant reconstruction) and (6.13), we infer that for any n =,..., N 1, any t (t (n), t (n+1) ], δ D β(u)(t)π D ζ(u (n+1) 1 ) = δt (n+ 1 2 ) 1 δt (n+ 1 2 ) ( ) β(π D u (n+1) ) β(π D u (n) ) ζ(π D u (n+1) ) ( ) η(π D u (n+1) ) η(π D u (n) ). Hence, taking v = (ζ(u () ), ζ(u (1) ),..., ζ(u (k) ),,..., ) X D, in (6.9), we find

190 t (k) η(π (1) D u(x, t(k) ))dx + η(π D u () (x))dx Estimates on the approximate solution 179 t (k) Λ(x) (1) D ζ(u)(x, t) (1) D ζ(u)(x, t)dxdt f(x, t)π (1) D ζ(u)(x, t)dxdt. (6.14) Equation (6.11) is a straightforward consequence of this estimate, of the relation Π (1) D u(, T ) = Π (1) D u(, t(k) ) (see (4.2)) and of the fact that the integrand involving Λ is nonnegative on [T, t (k) ]. Using the Young inequality (C.9), we write t (k) We also notice that f(x, t)π (1) D ζ(u)(x, t)dxdt C2 D 2λ f 2 L 2 ( (,t (k) )) + λ 2C 2 D s s 2 η(s) L β L ζ qdq = L β L ζ so that η(π (1) D u(, T )) = L 1 () and η(π D u () ) L 1 () Π (1) D ζ(u) 2 L 2 ( (,t (k) )). (6.15) η(π (1) D u(x, T ))dx = η(π D I D u ini ) L 1 () L βl ζ 2 2, (6.16) Π D I D u ini 2 L 2 (). The first two estimates in (6.12) therefore follow from (6.14), (6.15), Assumption (6.3e) on Λ, and the definition (2.1) of C D. Let us now prove that the uniform-in-time L 1 () estimate on Π (1) D η(u) implies the uniform-in-time L 2 () estimate on Π (1) D β(u) = β(π(1) D u). Owing to (6.3b), for all s there holds ζ(s) M s M 1 M L β β(s) M 1. Hence, using the Young inequality, η(s) = s ζ(q)β (q)dq M L β s s β(q)β (q)dq M 1 β (q)dq = M 2L β β(s) 2 M 1 β(s) M 2L β β(s) 2 M 4L β β(s) 2 L βm 2 1 M. For s, we use ζ(s) M s M 1 M L β β(s) M 1 to infer the same estimate. Therefore,

191 18 6 Degenerate parabolic problems s R, M 4L β β(s) 2 L βm 2 1 M η(s). (6.17) Making s = Π (1) D u in this inequality and using the uniform-in-time L1 () estimate on η(π (1) D u), we deduce the uniform-in-time L2 () estimate on β(π (1) D u) stated in (6.12). Corollary 6.11 (Existence of a solution to the GS). Under Assumptions (6.3), let D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Assume that the underlying spatial discretisation has a piecewise constant reconstruction in the sense of Definition 2.1. Then there exists at least a solution to the GS (6.9). Proof. For ρ [, 1] we let β ρ (u) = ρu + (1 ρ)β(u) and ζ ρ (u) = ρu + (1 ρ)ζ(u). It is clear that β ρ and ζ ρ satisfy the same assumptions as β and ζ for some L β and M, M 1 not depending on ρ. We can therefore apply Lemma 6.1 to see that there exists C 2, not depending on ρ, such that any solution u ρ to (6.9) with β = β ρ and ζ = ζ ρ satisfies (1) D ζ ρ(u ρ ) C 2. L 2 ((,T ) ) d Since D L 2 () d is a norm on X D,, this shows that (ζ ρ (u ρ )) ρ [,1] remains bounded in this finite dimensional space. In particular, for all i I, (ζ ρ (u ρ ) i ) ρ [,1] is bounded. Using Assumption (6.3b) for ζ ρ with constants not depending on ρ, we deduce that ((u ρ ) i ) ρ [,1] remains bounded for any i I, and thus that (u ρ ) ρ [,1] is bounded in X D,. If ρ = then (6.9) is a square linear system. Any solution to this system being bounded in X D,, this shows that the underlying linear system is invertible. A topological degree argument (see Theorem C.1) combined with the uniform bound on (u ρ ) ρ [,1] then shows that the scheme corresponding to ρ = 1, that is (6.9), possesses at least one solution. Lemma 6.12 (Uniqueness of the solution to the GS). Under Assumptions (6.3), let D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Assume that the underlying spatial discretisation has a piecewise constant reconstruction in the sense of Definition 2.1. Let u, ũ be solutions to the GS (6.9). Then, for all n =,..., N, Π D u (n) = Π D ũ (n) in L 2 (), and ζ(u (n) ) = ζ(ũ (n) ) in X D,. Proof. The proof is done by induction on n. The result is clearly true for n =, since u () = ũ () = I D u ini. Let us now assume that, for some n N 1, Π D u (n) (x) = Π D ũ (n) (x) for a.e. x. Subtracting the equation corresponding to ũ (n+1) to the equation corresponding to u (n+1), we get

192 6.3 Estimates on the approximate solution 181 [ ΠD (β(u (n+1) ) β(ũ (n+1) ))(x) Π δt (n+ 1 2 ) D v(x) ] + D (ζ(u (n+1) ) ζ(ũ (n+1) ))(x) D v(x) dx =, v X D,. (6.18) Using (6.3b)-(6.3c) we have ] [ ] Π D [β(u (n+1) ) β(ũ (n+1) ) Π D ζ(u (n+1) ) ζ(ũ (n+1) ) = [ ] [ ] β(π D u (n+1) ) β(π D ũ (n+1) ) ζ(π D u (n+1) ) ζ(π D ũ (n+1) )) Hence, making v = ζ(u (n+1) ) ζ(ũ (n+1) ) in (6.18), D (ζ(u (n+1) ) ζ(ũ (n+1) ))(x) 2 dx =.. Since L 2 () is a norm on X D,, this shows that ζ(u (n+1) ) = ζ(ũ (n+1) ). We then get, from (6.18), that [ ] Π D (β(u (n+1) ) β(ũ (n+1) ))(x) Π D v(x)dx =, v X D,. Letting v = β(u (n+1) ) β(ũ (n+1) ) gives Π D β(u (n+1) ) = Π D β(ũ (n+1) ) a.e. on. Since Π D ζ(u (n+1) ) = Π D ζ(ũ (n+1) ) a.e. on, Assumption (6.3d) and the fact that Π D (β(w) + ζ(w)) = β(π D w) + ζ(π D w) for all w X D, imply Π D u (n+1) = Π D ũ (n+1) a.e. on. Lemma 6.13 (Estimate on the dual norm of the discrete time derivative). Under Assumptions (6.3), let D T be a space time GD for homogeneous Dirichlet boundary conditions, in the sense of Definition 4.1. Assume that the underlying spatial discretisation has a piecewise constant reconstruction in the sense of Definition 2.1. Let u be a solution to Scheme (6.9). Then there exists C 3, depending only on L β, L ζ, C P C D, C ini Π D I D u ini L 2 (), f, λ, λ and T, such that T δ D β(u)(t) 2,D dt C 3, (6.19) where the dual norm,d is given by Definition Proof. Let us take a generic v = (v (n) ) n=1,...,n X D, as test function in (6.9). We have λ T T δ D β(u)(x, t)π (1) D v(x, t)dxdt T (1) D ζ(u)(x, t) (1) D v(x, t) dxdt + f(x, t)π (1) D v(x, t)dxdt.

193 182 6 Degenerate parabolic problems Using the Cauchy-Schwarz inequality, the definition 2.2 of C D, and Estimates (6.12), this gives C 4 > depending only on L β, C P, C ini, f, λ and λ such that T δ D β(u)(x, t)π (1) D v(x, t)dxdt C 4 (1) D v L 2 (,T ;L 2 ()) d. The proof of (6.19) is completed by selecting ( δ ) (n+ v = 1 2 ) β(u) z (n),d D n=,...,n with (z (n) ) n=,...,n X D, such that, for any n =,..., N 1, z (n+1) realises the supremum in (4.28) with w = δ (n+ 1 2 ) D β(u). 6.4 A first convergence theorem The following theorem states initial convergence properties of the GS for (6.1). Theorem 6.14 (Convergence of the GS). Under Assumptions (6.3), let ((D T ) m ) m N be a space time-consistent, limit-conforming and compact sequence of space time GDs, for homogeneous Dirichlet boundary conditions, in the sense of Definitions 4.3 and 4.6. We assume that the sequence of underlying spatial discretisations has the piecewise constant reconstruction property (Definition 2.1). Let ν : R R be defined by s R, ν(s) = s ζ (q)β (q)dq. (6.2) For any m N, let u m be a solution to (6.9) with D =. Then, as m, Π (1) β(u m ) β(u) weakly in L 2 () uniformly on [, T ] (see Definition 4.29), Π (1) ζ(u m ) ζ(u) weakly in L 2 ( (, T )), Π (1) ν(u m ) ν(u) in L 2 ( (, T )), (1) ζ(u m ) ζ(u) weakly in L 2 ( (, T )) d, (6.21) where u is the unique solution to (6.4). Remark We do not assume the existence of a solution u to the continuous problem, the convergence analysis establishes this existence, which proves part of Theorem 6.4.

194 6.4 A first convergence theorem 183 Proof. Note that, since ((D T ) m ) m N is compact, it is also coercive. Step 1: Application of compactness results. Thanks to Theorem 4.32 and Estimates (6.12) and (6.19), we first extract a subsequence, without changing the notations, such that (Π (1) β(u m )) m N converges weakly in L 2 () uniformly in [, T ] (in the sense of Definition 4.29) to some function β C([, T ]; L 2 ()-w). By space time-consistency of ((D T ) m ) m N, Π (1) β(u m )(, ) = Π Dm β(i Dm u ini ) = β(π Dm I Dm u ini ) β(u ini ) in L 2 (). Hence, the uniform-in-time weak L 2 () convergence of (Π (1) β(u m )) m N shows that β(, ) = β(u ini ) in L 2 (). Using again Estimates (6.12) and applying Lemma 4.7, we extract another subsequence such that, for some ζ L 2 (, T ; H 1 ()), Π (1) ζ(u m ) ζ weakly in L 2 ( (, T )) and (1) ζ(u m ) ζ weakly in L 2 ( (, T )) d. Estimates (6.12) and (6.19) also show that β m = β(u m ) and ζ m = ζ(u m ) satisfy the assumptions of Theorem 4.24 (weak-strong time-space convergence of a product theorem). Hence, T ( ) ( ) lim β Π (1) m u m (x, t) ζ Π (1) u m (x, t) dxdt T = β(x, t)ζ(x, t)dxdt. (6.22) Assumptions (6.3b) (6.3d) allow us to apply Lemma C.5 to w m = Π (1) u m. This gives the existence of a measurable function u such that β = β(u) and ζ = ζ(u) a.e. on (, T ). Since ζ L 2 ( (, T )), the growth assumption (6.3b) on ζ ensures that u L 2 ( (, T )). Since ζ (q)β (q) L ζ L β ζ (q)β (q), the following inequality holds for all a, b R: ( b (ν(a) ν(b)) 2 = ζ (q)β (q)dq It can therefore be deduced that T a ) 2 ( Lζ b L β β (q)ζ (q)dq a ) 2 ( ) ( b ) b L ζ L β β (q)dq ζ (q)dq a = L ζ L β [β(b) β(a)][ζ(b) ζ(a)]. [ ν(π (1) u m (x, t)) ν(u(x, t))] 2 dxdt a

195 184 6 Degenerate parabolic problems T L ζ L β [ ] β(π (1) u m (x, t)) β(u(x, t)) [ ] ζ(π (1) u m (x, t)) ζ(u(x, t)) dxdt. (6.23) Developing the right-hand side of this inequality, using (6.22) and the weak convergences β(π (1) u m ) β = β(u) and ζ(π (1) u m ) ζ = ζ(u), we see that this right-hand side goes to as m. Hence, taking the superior limit as m in (6.23) shows that ν(π (1) u m ) ν(u) in L 2 ( (, T )). Step 2: u is a solution to (6.4). We drop some indices m for legibility. Let v L 2 (, T ; H 1 ()) such that t v L 2 ( (, T )) and v(, T ) =. Let (v m ) m N be given by Lemma 4.9 (for θ = ) and introduce (, v (),..., v (N 1) ) as test function in (6.9). This gives T (m) 1 + T (m) 2 = T (m) 3 with and T (m) 1 = N 1 n= T (m) 2 = [ ] Π D β(u (n+1) )(x) Π D β(u (n) )(x) Π D v (n) (x)dx, T Λ(x) (1) D T T (m) 3 = ζ(u)(x, t) () D v(x, t)dxdt, f(x, t)π () D v(x, t)dxdt. Use the discrete integration-by-parts formula (C.15) in T (m) T (m) T 1 = 1 : Π (1) D β(u)(x, t)δ Dv(x, t)dxdt β(π D I D u ini )(x)π D v () dx. Hence, by the convergence properties (4.1c) and (4.1b) of (v m ) m N, as m we have T T (m) 1 β(u)(x, t) t v(x, t)dxdt β(u ini )(x)v(x, )dx. (6.24) Using the convergence (4.7b) and (4.7a) of (v m ) m N, we have, as m, T T (m) 2 Λ(x) ζ(u)(x, t) v(x, t)dxdt, T T (m) 3 f(x, t)v(x, t)dxdt. Plugged alongside (6.24) in T (m) 1 + T (m) 2 = T (m) 3, these convergences show that u satisfies (6.4).

196 6.4 A first convergence theorem 185 Remark 6.16 (Convergence of Π (1) D um?) We do not prove here that u is a weak limit of Π (1) u m. Such a limit is not stated in (6.21) and can actually be considered as irrelevant for the model (6.1) since, in this model, the quantities of interest (physically relevant when this PDE models a natural phenomenon) are β(u) and ζ(u). As a corollary to this convergence analysis and to the uniqueness of the solution (Theorem 6.4), an equivalent form of (6.4) can be stated. Corollary 6.17 (Equivalent from of (6.4)). Under Hypotheses (6.3), Problem (6.4) is equivalent to u L 2 (, T ; L 2 ()), ζ(u) L 2 (, T ; H 1 ()), β(u) C([, T ], L 2 ()-w), t β(u) L 2 (, T ; H 1 ()), β(u)(, ) = β(u ini ) in L 2 (), T t β(u)(, t), v(, t) H 1,H 1dt T + Λ(x) ζ(u)(x, t) v(x, t)dxdt T = f(x, t)v(x, t)dxdt, v L 2 (; T ; H 1 ()), (6.25) where C([, T ]; L 2 ()-w) denotes the space of continuous functions [, T ] L 2 () for the weak- topology of L 2 (). Proof. Let us prove that (6.4) implies (6.25). There is a unique solution to (6.4) (Theorem 6.4), so it must be the u constructed in the proof of Theorem We saw in Step 1 of this proof that β(u) = β C([, T ]; L 2 ()-w) and that β(u)(, ) = β(, ) = β(u ini ). Using Cc ((, T ) ) test functions in (6.4), we see that t β(u) = div(λ ζ(u)) + f in the sense of distributions. Since ζ(u) L 2 (, T ; L 2 ()), this shows that t β(u) L 2 (, T ; H 1 ()). Let v Cc ((, T ) ). By definition of the distribution derivative, T β(u)(x, t) t v(x, t)dxdt = T t β(u)(t), v(t) H 1,H 1 dt and thus (6.4) shows that the equation in (6.25) is satisfied for such smooth compactly supported v. Since these functions are dense in L 2 (, T ; H 1 ()) (see [31]), we infer that (6.25) is fully satisfied. Let us now assume that u satisfies (6.25). Then it clearly has all the regularity properties expected in (6.4). To prove that it also satisfies the equation in this latter problem, we start by taking v Cc ((, T ) ). By smoothness

197 186 6 Degenerate parabolic problems of this function and regularity assumptions on β(u) an integration-by-parts gives T T t β(u)(t), v(t) H 1,H 1dt = β(u)(x, t) t v(x, t)dxdt β(u)(x, )v(x, )dx. Since β(u)(x, ) = β(u ini ), (6.25) proves that (6.4) is satisfied for such v. As discussed at the end of the proof of Theorem 6.14, this shows that (6.25) is satisfied for all required test functions. Remark 6.18 (The continuity property of β(u)) The continuity property of β(u) : [, T ] L 2 ()-w is rather natural. Indeed, the PDE in the sense of distributions shows that T ϕ : t β(u)(t), ϕ L 2 belongs to W 1,1 (, T ), and is therefore continuous, for any ϕ Cc (). The density in L 2 () of such ϕ, combined with the fact that β(u) L (, T ; L 2 ()), proves the continuity of T ϕ for any ϕ L 2 (), that is to say the continuity of β(u) : [, T ] L 2 ()-w. This notion of β(u) as a function continuous in time is nevertheless a subtle one. It is to be understood in the sense that the function (x, t) β(u(x, t)) has an a.e. representative which is continuous [, T ] L 2 ()-w. In other words, there is a function Z C([, T ]; L 2 ()-w) such that Z(t)(x) = β(u(x, t)) for a.e. (x, t) (, T ). We must however make sure, when dealing with pointwise values in time to separate Z from β(u(, )) as β(u(, t 1)) may not make sense for a particular t 1 [, T ]. That being said, in order to adopt a simple notation, in the following we denote by β(u)(, ) the function Z, and by β(u(, )) the a.e.-defined composition of β and u. Hence, it will make sense to talk about β(u)(, t) for a particular t 1 [, T ], and we will only write β(u)(x, t) = β(u(x, t)) for a.e. (x, t) (, T ). 6.5 Uniform-in-time, strong L 2 convergence results We denote by R β the range of β and define the pseudo-inverse function β i : R β R of β by { s R β, β i inf{t R β(t) = s} if s, (s) = sup{t R β(t) = s} if s <, = closest t to such that β(t) = s. (6.26) See Figure 6.1 for an illustration of β i. Since β() =, it holds β i on R β R + and β i on R β R. The function B : R β [, ] is defined by B(z) = z ζ(β i (s)) ds. (6.27)

198 6.5 Uniform-in-time, strong L 2 convergence results 187 β i y = x b β a b a Fig An example of β (dashed line) and its pseudo-inverse function β i (continuous line). Here, the range of β is [a, b). The function β i is non-decreasing, and thus B(z) is always well-defined in [, ). The signs of β i and ζ also ensure that that B is non-decreasing on R β R + and non-increasing on R β R. B can therefore be extended to the closure R β of R β, by defining B(a) = lim z a B(z) [, + ] at any endpoint a of R β that does not belong to R β. Lemma 6.23 in Section 6.6 states a few useful properties of B. Remark 6.19 (Range of β(u)) The a.e. equality β(u)(x, t) = β(u(x, t)) (see Remark 6.18) ensures that β(u)(, ) takes its values in R β. The following theorem shows that the solutions to GSs for (6.1) actually enjoy stronger convergence results that established in Theorem Theorem 6.2 (Uniform-in-time convergence of the GS). Under the assumptions of Theorem 6.14, the solution u m to the GS (6.9) with D T = (D T ) m satisfies the following convergence results, as m :

199 188 6 Degenerate parabolic problems sup Π (1) ν(u m )(t) ν(u)(t), t [,T ] L2 () Π (1) ζ(u m ) ζ(u) in L 2 ( (, T )), (1) ζ(u m ) ζ(u) in L 2 ( (, T )) d, (6.28) where u is the unique solution to (6.4). Remark For the Stefan model, β = Id and thus ν(u) = ζ(u) is the temperature of the melting material. For the Richards model, ζ = Id and thus ν(u) = β(u) is the water saturation. Hence, in both cases, ν(u) is the quantity of interest to approximate. Proof. By (6.38) in Lemma 6.23, η = B β. The energy estimate (6.11) can thus be written B(β(Π (1) u m ))(x, T )dx + T Λ(x) (1) ζ(u m )(x, t) (1) ζ(u m )(x, t)dxdt B(β(Π Dm I Dm u ini ))(x)dx t (k) + f(x, t)π (1) ζ(u m )(x, t)dxdt. (6.29) Here, we recall that t (k) is the time such that T (t (k 1), t (k) ]. Step 1 Uniform-in-time convergence of Π (1) ν(u m ). Let us take T [, T ] and (T m ) m 1 a sequence in [, T ] which converges to T. The Cauchy Schwarz inequality for the semi-definite positive symmetric form Tm W L 2 ((, T ) ) d Λ(x)W (t, x) W (t, x)dxdt shows that ( Tm Λ(x) (1) ζ(u m )(t, x) ζ(u)(t, x)dxdt ( Tm ) 2 Λ(x) (1) ζ(u m )(t, x) (1) ζ(u m )(t, x)dxdt ( Tm ) Λ(x) ζ(u)(t, x) ζ(u)(t, x)dxdt )

200 6.5 Uniform-in-time, strong L 2 convergence results 189 By weak convergence in L 2 ((, T ) ) d of (1) ζ(u m ) to ζ(u) and strong convergence in the same space of 1 [,Tm] ζ(u) to ζ(u), we can pass to the limit in the left-hand side (by weak-strong convergence, see Lemma C.3 page 45) and in the second term in the right-hand side. Hence, taking the inferior limit of this inequality and dividing by T Λ ζ(u) ζ(u), we deduce that Tm lim inf Λ(x) (1) m ζ(u m )(x, t) (1) ζ(u m )(x, t)dxdt T Λ(x) ζ(u)(x, t) ζ(u)(x, t)dxdt. (6.3) By space time-consistency of ((D T ) m ) m N growth (6.16) of η = B β, it holds (Definition 4.3) and quadratic B(β(Π Dm I Dm u ini )) B(β(u ini ) in L 1 () as m. (6.31) We then write (6.29) with T m instead of T. The time t (k(m)) such T m (t (k(m) 1), t (k(m)) ] satisfies t (k(m)) T as m. Hence, using (6.3) and the weak convergence of Π (1) ζ(u m ) to ζ(u), lim sup B(β(Π (1) u m (x, T m )))dx m ( lim sup m + t (k(m)) Tm B(β(Π Dm I Dm u ini ))(x)dx f(x, t)π (1) ζ(u m )(x, t)dxdt B(β(u ini ))(x)dx + Tm Λ(x) (1) ζ(u m )(x, t) (1) ζ(u m )(x, t)dxdt T f(x, t)ζ(u)(x, t)dxdt lim inf Λ(x) Dm ζ(u m )(x, t) Dm ζ(u m )(x, t)dxdt m T B(β(u ini ))(x)dx + f(x, t)ζ(u)(x, t)dxdt T Λ(x) ζ(u)(x, t) ζ(u)(x, t)dxdt. Corollary 6.26 therefore gives lim sup B(β(Π (1) u m (x, T m )))dx m ) B(β(u)(x, T ))dx. (6.32)

201 19 6 Degenerate parabolic problems By Lemma 4.28, the uniform-in-time weak L 2 convergence of β(π (1) u m ) to β(u) and the continuity of β(u) : [, T ] L 2 ()-w (see Corollary 6.17), we have β(π (1) u m )(T m ) β(u)(t ) weakly in L 2 () as m. (6.33) Therefore, for any (s m ) m N converging to T, β(π (1) u m (T m )) + β(u)(s m ) β(u)(t ) weakly in L 2 () as m. 2 Lemma C.6 then gives, by convexity of B, B(β(u)(x, T ))dx ( ) (1) β(π D lim inf B m u m (x, T m )) + β(u)(x, s m ) dx. (6.34) m 2 Property (6.4) of B and the two inequalities (6.32) and (6.34) allow us to conclude the proof. Let T be the set of τ [, T ] such that β(u(, τ)) = β(u)(, τ) and ν(u(, τ)) = ν(u)(, τ) a.e. on (see Remarks 6.18 and 6.27), and let (s m ) m N be a sequence in T which converges to T. Since ν(u) C([, T ]; L 2 ()) by Corollary 6.26, we have Inequality (6.4) gives ν(u(, s m )) ν(u)(, T ) in L 2 () as m. (6.35) ν(π (1) u m (, T m )) ν(u)(, T ) 2 L 2 () 2 ν(π (1) u m (, T m )) ν(u(, s m )) 2 L 2 () + 2 ν(u(, s m )) ν(u)(, T )) 2 L 2 () [ ] 8L β L ζ B(β(Π (1) u m (x, T m ))) + B(β(u(x, s m ))) dx ( ) (1) β(π D 16L β L ζ B m u m (x, T m )) + β(u(x, s m )) dx ν(u(, s m )) ν(u)(, T )) 2 L 2 (). We then take the lim sup as m of this expression. Thanks to (6.32) and to the boundedness of B : t [, T ] B(β(u)(x, t))dx [, ) (see Corollary 6.26), the first term in the right-hand side has a finite lim sup. We can therefore split the lim sup of this right-hand side without risking writing

202 6.5 Uniform-in-time, strong L 2 convergence results 191 and we get, thanks to (6.32), (6.34), (6.35) and to the continuity of B (Corollary 6.26), lim sup ν(π (1) u m (, T m )) ν(u)(, T ) 2. m L 2 () Thus, ν(π (1) u m (, T m )) ν(u)(t ) strongly in L 2 (). By Lemma 4.28, this concludes the proof that sup t [,T ] ν(π (1) u m )(t) ν(u)(t) L2 (). Step 2: Strong convergence of (1) ζ(u m ). Since B is convex, the convergence property (6.33) (with T m = T = T ) and Lemma C.6 give B(β(u)(x, T ))dx lim inf m B(β(Π (1) u m )(x, T ))dx. Writing (6.29) with T = T, taking the lim sup as m, using (6.31) and the continuous integration-by-part formula (6.49), we therefore find T lim sup Λ(x) (1) ζ(u m )(x, t) (1) ζ(u m )(x, t)dxdt m T Λ(x) ζ(u)(x, t) ζ(u)(x, t)dxdt. Combined with (6.3) with T m = T = T, this shows that T lim Λ(x) (1) m ζ(u m )(x, t) (1) ζ(u m )(x, t)dxdt T = Λ(x) ζ(u)(x, t) ζ(u)(x, t)dxdt. (6.36) Developing all the terms and using the weak convergence of (1) ζ(u m ) to ζ(u), we deduce T [ lim Λ(x) (1) m ] ζ(u m )(x, t) ζ(u)(x, t) [ (1) ζ(u m )(x, t) ζ(u)(x, t) ] dxdt =. The coercivity of Λ therefore implies (1) ζ(u m ) ζ(u) strongly in L 2 ( (, T )) d as m. Step 3: Strong convergence of Π (1) ζ(u m ). Apply Lemma 4.9 to v = ζ(u). This gives (v m ) m N such that Π (1) v m ζ(u) in L 2 ( (, T )) and (1) v m ζ(u) in L 2 ( (, T )) d. The coercivity definition 2.2 gives

203 192 6 Degenerate parabolic problems Π (1) ζ(u m ) Π (1) L2 v m ( (,T )) C P (1) ζ(u m ) (1) v m L 2 ( (,T )) d. By strong convergence of (1) ζ(u m ), letting m in this estimate proves that Π (1) ζ(u m ) ζ(u) in L 2 ( (, T )) follows. Remark 6.22 (Convergence of B(β(Π (1) u m(t m)))) Let T m T. The convergence property (6.33), the convexity of B and Lemma C.6 show that B(β(u)(x, T ))dx lim inf B(β(Π (1) m u m)(x, T m))dx. Combined with (6.32), this gives lim m B(β(Π (1) u m(x, T m)))dx = B(β(u)(x, T ))dx. (6.37) 6.6 Auxiliary results We state here a family of technical lemmas, starting with a few properties on ν and B. Lemma Under Assumptions (6.3b) and (6.3c), let ν be defined by (6.2), B be defined by (6.27), and η be defined by (6.1). Then the function B is convex lower semi-continuous on R β, the function B β : R [, ) is continuous, and s R, η(s) = B(β(s)) = s ζ(q)β (q)dq, (6.38) a R, r R β, B(r) B(β(a)) ζ(a)(r β(a)), (6.39) s, s R, (ν(s) ν(s )) 2 4L β L ζ [B(β(s)) + B(β(s )) ( β(s) + β(s ) ] ) 2B. (6.4) 2

204 6.6 Auxiliary results 193 Proof. Let us first notice that, since β on R + and β on R, β i (s) is a real number for all s R β. Moreover, since β is non-decreasing, β i is also non-decreasing on R β and therefore locally bounded on R β. Hence, B is well defined and locally Lipschitz-continuous, with an a.e. derivative B = ζ(β i ). B is therefore non-decreasing and B is convex. Since B is continuous on R β and extended by its (possibly infinite) limit at the endpoints of this interval, B is lower semi-continuous on R β. To prove (6.38), we denote by P R β the countable set of plateaux values of β, i.e. the numbers y R such that β 1 ({y}) is not reduced to a singleton. If s β 1 (P ) then β 1 ({β(s)}) is the singleton {s} and therefore β i (β(s)) = s. Moreover, β i is continuous at β(s) and thus B is differentiable at β(s). Since β is differentiable a.e., we deduce that, for a.e. s β 1 (P ), (B(β)) (s) = B (β(s))β (s) = ζ(β i (β(s)))β (s) = ζ(s)β (s). The set β 1 (P ) is a union of intervals on which β, and thus B(β), are locally constant; hence, for a.e. s in this set, (B(β)) (s) = and ζ(s)β (s) =. As a consequence, the locally Lipschitz-continuous functions B(β) and s s ζ(q)β (q)dq have identical derivatives a.e. on R. Since they have the same value at s =, they are thus equal on R and the proof of (6.38) is complete. The continuity of B β follows from this relation. We now prove (6.39), which states that ζ(a) belongs to the convex subdifferential of B at β(a). We first start with the case r R β, that is r = β(b) for some b R. If β i is continuous at β(a) then B is differentiable at β(a), with B (β(a)) = ζ(β i (β(a))) = ζ(a), and (6.39) is an obvious consequence of the convexity of B. Otherwise, a plain reasoning also does the job as B(r) B(β(a)) = B(β(b)) B(β(a)) = = b a b a ζ(q)β (q)dq (ζ(q) ζ(a))β (q)dq + ζ(a)(β(b) β(a)) ζ(a)(r β(a)). Here, the inequality comes from the fact that β and that ζ(q) ζ(a) has the same sign as b a if q is between a and b. The general case r R β is obtained by passing to the limit on b n such that β(b n ) r, and by using the fact that B has limits (possibly + ) at the endpoints of R β. Let us now take s, s R, and let s R be such that β( s) = β(s)+β(s ) 2. We have s s β (q)dq + β (q)dq = β(s) + β(s ) 2β( s) =. s s Hence, using (6.38), B(β(s))+B(β(s )) 2B(β( s))

205 194 6 Degenerate parabolic problems s s s = ζ(q)β (q)dq + ζ(q)β (q)dq 2 ζ(q)β (q)dq s s = ζ(q)β (q)dq + ζ(q)β (q)dq s s s s = (ζ(q) ζ( s))β (q)dq + (ζ(q) ζ( s))β (q)dq. (6.41) s s We then use ζ(q) ζ( s) 1 L β ν(q) ν( s) and β (q) β (q) ζ (q) L ζ write s s s (ζ(q) ζ( s))β (q)dq 1 ν (q)(ν(q) ν( s))dq L β L ζ s 1 = (ν(s) ν( s)) 2. 2L β L ζ The same relation holds with s replaced by s. Owing to (ν(s) ν(s )) 2 2(ν(s) ν( s)) 2 + 2(ν(s ) ν( s)) 2, the inequality (6.4) follows from (6.41). = ν (q) L ζ The following property states an expected integration-by-parts result, which can be formally obained by writing ( t β(v))ζ(v) = β (v)ζ(v) t v = t B(β(v)) (owing to (6.38)). The rigorous proof of this result is however a bit technical, due to the lack of regularity on u and to the non-linearities involved. Lemma Let us assume (6.3b) and (6.3c). Let v : (, T ) R be measurable such that ζ(v) L 2 (, T ; H 1 ()), B(β(v)) L (, T ; L 1 ()), β(v) C([, T ]; L 2 ()-w), t β(v) L 2 (, T ; H 1 ()). Then t [, T ] B(β(v)(x, t))dx [, ) is continuous and, for all t 1, t 2 [, T ], to t2 t 1 t β(v)(t), ζ(v(t)) H 1,H 1dt = B(β(v)(x, t 2 ))dx B(β(v)(x, t 1 ))dx. (6.42) Remark 6.25 (Continuity of β(v)) Since η = B β satisfies (6.17), the condition B(β(v)) L (, T ; L 1 ()) ensures that β(v) L (, T ; L 2 ()). Combined with the condition tβ(v) L 2 (, T ; H 1 ()), this shows that β(v) C([, T ]; L 2 ()-w). Hence, this continuity property on β(v) is actually a consequence of the other assumptions on v. We also point out that, as in Remark 6.18, it is important to keep in mind the separation between β(v(, )) and its continuous representative β(v)(, ).

206 Proof. We obviously only need to make the proof when t 1 < t 2. Step 1: truncation, extension and approximation of β(v). We define β(v) : R L 2 () by setting β(v)(t) if t [t 1, t 2 ], β(v)(t) = β(v)(t 1 ) if t t 1, β(v)(t 2 ) if t t Auxiliary results 195 By continuity property of β(v), we have β(v) C(R; L 2 ()-w) and t β(v) = 1 (t1,t 2) t β(v) L 2 (R; H 1 ()) (no Dirac masses have been introduced at t = t 1 or t = t 2 ). This regularity of t β(v) ensures that the function t R D h β(v) := 1 h = t+h t t β(v)(s)ds (6.43) β(v)(t + h) β(v)(t) h tend to t β(v) in L 2 (R; H 1 ()) as h. H 1 () (6.44) Step 2: we prove that B(β(v)(t) L 1 () B(β(v)) L (,T ;L 1 ()) for all t R (not only for a.e. t). Let t [t 1, t 2 ]. Since β(v)(, ) = β(v(, )) a.e. on (t 1, t 2 ), there exists a sequence t n t such that, for all n, β(v)(, t n ) = β(v(, t n )) in L 2 () and B(β(v)(, t n )) L 1 () B(β(v)) L (,T ;L 1 ()). Using the continuity of β(v) with values in L 2 ()-w, we have β(v)(, t n ) β(v)(, t) weakly in L 2 (). We then use the convexity of B and Lemma C.6 to write, thanks to our choice of t n, B(β(v)(x, t))dx lim inf B(β(v)(x, t n ))dx B(β(v)) n L (,T ;L 1 ()). The estimate on B(β(v))(t) is thus complete for t [t 1, t 2 ]. The result for t t 1 or t t 2 is obvious since β(v)(t) is then either β(v)(t 1 ) or β(v)(t 2 ). Step 3: We prove that for all τ R and a.e. t (t 1, t 2 ), β(v)(τ) β(v)(t), ζ(v(, t)) H 1,H 1 B(β(v)(x, τ)) B(β(v)(x, t))dx. (6.45) Let t such that β(v)(, t) = β(v(, t)) a.e. on. Almost every t satisfies this property. By Remark 6.19, for a.e. x we have β(v)(x, τ) R β and we can therefore write, by (6.39) with r = β(v)(x, τ) and a = v(x, t), B(β(v)(x, τ)) B(β(v(x, t))) ζ(v(x, t))(β(v)(x, τ) β(v(x, t))).

207 196 6 Degenerate parabolic problems Integrating this relation over x, Property (6.45) follows since the H 1 H 1 duality product in (6.45) can be replaced with an L 2 inner product, as all terms in this product belong to L 2 (). Step 4: proof of (6.42) By convergence of D h β(v) to t β(v) in L 2 (, T ; H 1 ()) and since 1 (t1,t 2)ζ(v) L 2 (R; H 1 ()), we have t2 t 1 t β(v)(t), ζ(v(t)) H 1,H 1dt = t β(v)(t), 1 (t1,t 2)(t)ζ(v(, t)) H 1,H 1dt R = lim D h β(v)(t), 1 (t1,t h 2)(t)ζ(v(, t)) H 1,H 1dt 1 = lim h h R t2 β(v)(s + h) β(v)(t), ζ(v(, t) H 1,H 1 dt. (6.46) t 1 We then use (6.45) for a.e. t (t 1, t 2 ) to obtain, for h small enough such that t 1 + h < t 2, 1 h t2 t 1 β(v)(t + h) β(v)(t), ζ(v(, t)) H 1,H 1 dt 1 t2 B(β(v)(x, t + h)) B(β(v)(x, t))dxdt h t 1 = 1 t2+h B(β(v)(x, t))dxdt 1 t1+h B(β(v)(x, t))dxdt h t 2 h t 1 = B(β(v)(x, t 2 ))dx 1 t1+h B(β(v)(x, t))dxdt. (6.47) h t 1 In the last line, we used β(v)(t) = β(v)(t 2 ) for all t t 2. We then take the superior limit of (6.47), and use the fact that B(β(v)(, t 2 )) is integrable (Step 2) to take its integral out of the lim sup. Coming back to (6.46) we obtain t2 t 1 t β(v)(t), ζ(v(t)) H 1,H 1 dt B(β(v)(x, t 2 ))dx lim inf h 1 h t1+h t 1 B(β(v)(x, t))dxdt. (6.48) t1+h But since β(v) C([, T ]; L 2 ()-w), as h we have 1 h t 1 β(v)(t)dt β(v)(t 1 ) weakly in L 2 (). Hence, the convexity of B, Lemma C.6 and Jensen s inequality give ( ) 1 t1+h B(β(v)(x, t 1 ))dx lim inf B β(v)(x, t)dt dx h h t 1

208 1 lim inf h h t1+h t Auxiliary results 197 B(β(v)(x, t))dtdx. Plugged into (6.48), this inequality shows that (6.42) holds with instead of =. The reverse inequality is obtained by reversing time. We consider ṽ(t) = v(t 1 + t 2 t). Then ζ(ṽ), B(β(ṽ)) and β(ṽ) have the same properties as ζ(v), B(β(v)) and β(v), and β(ṽ) takes values β(v)(t 1 ) at t = t 2 and β(v)(t 2 ) at t = t 1. Applying (6.42) with instead of = to ṽ and using the fact that t β(ṽ)(t) = t β(v)(t 1 + t 2 t), we obtain (6.42) with instead of = and the proof of (6.42) is complete. The continuity of t [, T ] B(β(v)(x, t))dx is straightforward from (6.42), since the left-hand side of this relation is continuous with respect to t 1 and t 2. The following corollary states continuity properties and an essential formula on the solution to (6.4). Corollary Under Assumption (6.3), if u is a solution of (6.4) then: 1. The function t [, T ] B(β(u)(x, t))dx [, ) is continuous (and thus bounded); 2. For any T [, T ], T B(β(u)(x, T ))dx + = B(β(u ini (x)))dx + 3. ν(u) is continuous [, T ] L 2 (). Λ(x) ζ(u)(x, t) ζ(u)(x, t)dxdt T f(x, t)ζ(u)(x, t)dxdt; (6.49) Remark 6.27 (Continuity of ν(u)) The continuity of ν(u) has to be understood in the same sense as the continuity of β(u) (see Remark 6.18), that is, ν(u) is a.e. on (, T ) equal to a continuous function [, T ] L 2 (). We use in particular a similar notation ν(u)(, ) for the continuous representative of ν(u(, )) as we did for the continuous representative of β(u). Proof. We first notice that Corollary 6.17 was established using Theorems 6.4 and 6.14, which do not make use of Corollary Hence, we invoke Corollary 6.17, which tells us that u is also a solution to (6.25). The continuity of t [, T ] B(β(u)(x, t))dx [, ) and Formula (6.49) therefore follow from Lemma 6.24 applied to v = u, by using v = ζ(u)1 [,T] in (6.25). Let us prove the strong continuity of ν(u) : [, T ] L 2 (). Let T be the set of τ [, T ] such that β(u(, τ)) = β(u)(, τ) a.e. on. The set [, T ]\T has

209 198 6 Degenerate parabolic problems zero measure. Let (s l ) l N and (t k ) k N be two sequences in T that converge to the same value s. Owing to (6.4), [ν(u(x, s l )) ν(u(x, t k ))] 2 dx ( ) 4L β L ζ B(β(u(x, s l )))dx + B(β(u(x, t k )))dx ( ) β(u(x, sl )) + β(u(x, t k )) 8L β L ζ B dx (6.5) 2 ( ) = 4L β L ζ B(β(u)(x, s l ))dx + B[β(u)(x, t k ))dx ( ) β(u)(x, sl ) + β(u)(x, t k ) 8L β L ζ B dx. 2 Since β(u)(,s l)+β(u)(,t k ) 2 β(u)(, s) weakly in L 2 () as l, k, Lemma C.6 and the convexity of B (Lemma 6.23) give B (β(u)(x, s)) dx lim inf B l,k ( β(u)(x, sl ) + β(u)(x, t k ) 2 ) dx. Taking the superior limit as l, k of (6.5) and using the continuity of t B(β(u)(x, t))dx thus shows that ν(u(, s l )) ν(u(, t k )) L 2 () as l, k. (6.51) The existence of an a.e. representative of ν(u(, )) that is continuous [, T ] L 2 () is a direct consequence of this convergence. Let s [, T ] and (s l ) l N T that converges to s. Applied with t k = s k, (6.51) shows that (ν(u(, s l ))) l N is a Cauchy sequence in L 2 (), and therefore that lim l ν(u(, s l )) exists in L 2 (). Relation (6.51) also shows that this limit, that we can call ν(u)(, s), does not depend on the Cauchy sequence in T which converges to s. With t k = s, we also see that whenever s T we have ν(u(, s)) = ν(u)(, s) a.e. on, and ν(u)(, ) is therefore equal to ν(u(, )) a.e. on (, T ). It remains to establish that ν(u) thus defined is continuous [, T ] L 2 (). For any (τ r ) r N [, T ] which converges to τ [, T ], we can pick s r T (τ r 1 r, τ r + 1 r ) and t r T (τ 1 r, τ + 1 r ) such that We therefore have ν(u)(, τ r ) ν(u(, s r )) L2 () 1 r, ν(u)(, τ) ν(u(, t r )) L 2 () 1 r.

210 6.7 Proof of the uniqueness of the solution to the model 199 sup ν(u)(, τ r ) ν(u)(, τ) L 2 () t [,T ] 2 r + sup ν(u(, s r )) ν(u(, t r )) L 2 (). t [,T ] By (6.51) with l = k = r, this proves that ν(u)(, τ r ) ν(u)(, τ) in L 2 () as r. 6.7 Proof of the uniqueness of the solution to the model We give here a proof of the uniqueness of the solution to (6.4) (and thus also to the solution to (6.6)). The uniqueness of entropy solutions to t β(u) ζ(u) = f (with an additional convective term, and a merely integrable f) has been established in [17], using the doubling variable technique. Although this proof could be extended to our framework, we rather provide here a much shorter proof, following the idea due to J. Hadamard [62]. This idea consists in using the solution to an approximate dual problem. It was successfully applied to the one-dimensional Stefan problem in [9], and subsequently generalised to the higher dimensional case in [61]. The proof provided here was originally developed in [4] and applies the approximate duality technique to the doubly degenerate model (6.1), which contains both Richards and Stefan s models as particular case. Proof of uniqueness of the solution to (6.4). Set u d = β(u 1 ) + ζ(u 1 ) β(u 2 ) ζ(u 2 ), and for all (x, t) [, T ], define { ζ(u1(x,t)) ζ(u 2(x,t)) q(x, t) = u d (x,t) if u d (x, t), otherwise. Take ψ L 2 (, T ; H 1 ()) with t ψ L 2 ( (, T )), ψ(, T ) = and div(λ ψ) L 2 ( (, T )). Subtract the two equations (6.4) satisfied by u 1 and u 2, and use ψ as a test function. The assumed regularity div(λ ψ) L 2 ( (, T )) enables us to integrate by parts the term involving Λ u ψ, and we obtain T ( ) u d (x, t) (1 q(x, t)) t ψ(x, t) + q(x, t)div(λ ψ)(x, t) dxdt =. (6.52) For ε (, 1/2) set q ε = (1 2ε)q + ε. Since q 1 we have ε q ε 1 ε, and (q ε q) 2 (q ε q) 2 ε and ε. (6.53) q ε 1 q ε Let ψ ε be given by Lemma 6.28 below, with g = q ε and some w C c ( (, T )). Making ψ = ψ ε in (6.52) and using (6.56),

211 2 6 Degenerate parabolic problems T u d (x, t)w(x, t)dxdt T u d (x, t)(q ε (x, t) q(x, t))(div(λ ψ ε )(x, t) t ψ ε (x, t))dxdt. (6.54) The Cauchy-Schwarz inequality, (6.57) and (6.53) imply [ T u d (x, t)(q ε (x, t) q(x, t))(div(λ ψ ε )(x, t) t ψ ε (x, t))dxdt ( T ) 2 u d (x, t) 2 (q(x, t) q ε(x, t)) 2 dxdt q ε (x, t) ( T ) ( ) 2dxdt q ε (x, t) div(λ ψ ε )(x, t) ( T ) + 2 u d (x, t) 2 (q(x, t) q ε(x, t)) 2 dxdt 1 q ε (x, t) ( T ) ( ) 2dxdt (1 q ε (x, t)) t ψ ε (x, t) 2εC u d L2 ( (,T )) ( ) w 2 L 2 ( (,T )) + d w 2 L 2 ( (,T )) + tw 2 L 2 ( (,T )). (6.55) Letting ε and using (6.54) gives T u d (x, t)w(x, t)dxdt =. Since this holds for any function w C c ( (, T )), we deduce that u d = a.e. on (, T ). Hence β(u 1 ) + ζ(u 1 ) = β(u 2 ) + ζ(u 2 ), and the proof is complete since β + ζ is one-to-one. The following lemma ensures the existence of the function ψ, used in the proof above. Lemma Let T >, and let be a bounded open subset of R d (d N). Assume Hypothesis (6.3e). Let w Cc ( (, T )) and g L ( (, T )) such that g(x, t) [g min, 1 g min ] for a.e. (x, t) (, T ), where g min is a fixed number in (, 1 2 ). Then there exists a function ψ such that: 1. ψ L (, T ; H 1 ()), t ψ L 2 ( (, T )), div(λ ψ) L 2 ( (, T )) (this implies ψ C([, T ]; L 2 ())); 2. ψ(, T ) = ; ] 2

212 3. For a.e. (x, t) (, T ), 6.7 Proof of the uniqueness of the solution to the model 21 (1 g(x, t)) t ψ(x, t) + g(x, t)div(λ ψ)(x, t) = w(x, t); (6.56) 4. There exists C >, depending only on T, diam(), λ and λ (and not on g min ), such that Proof. T ( ( ) 2 ( ) ) 2 (1 g(x, t)) t ψ(x, t) + g(x, t) div(λ ψ)(x, t) dxdt ) C ( w 2 L 2 ( (,T )) + d w 2 L 2 ( (,T )) + tw 2 L 2 ( (,T )). (6.57) Step 1: existence of ψ satisfying 1, 2 and 3. After dividing through by g, observe that (6.56) is equivalent to For a.e. (x, t) (, T ), Φ(x, t) t ψ(x, t) + div(λ(x) ψ(x, t)) = f(x, t), (6.58) where f L ( (, T )), Φ L ( (, T )) and, for some fixed numbers ϕ ϕ >, ϕ Φ(x, t) ϕ for a.e. (x, t) (, T ). The parabolic equation (6.58) is slightly non-standard because of the time-dependent coefficient Φ in front of t ψ. However, as we shall now see, a standard Galerkin approximation provides the existence of a solution to this equation. Let (V k ) k N be a non-decreasing family of finite-dimensional subspaces of H 1 (). We look for ψ k : [, T ] V k solution to the following Galerkin approximation of (6.58), with final condition: ψ k (T ) = and t [, T ], v V k, (Φ(, t)ψ k(t), v) L 2 (Λ ψ k (t), v) (L 2 ) d = (f(, t), v) L 2. (6.59) Here, (, ) L 2 is the L 2 () inner product. Choosing an orthonormal (for this inner product) basis (e i ) i=1,...,nk of V k and writing ψ k (t) = N k i=1 θ i(t)e i, (6.59) can be re-cast as Θ(T ) = and, for all t [, T ], M(t)Θ (t) S(t)Θ(t) = F (t) (6.6) where Θ(t) = (θ i (t)) i=1,...,nk, M(t) and S(t) are the symmetric matrices with respective entries M i,j (t) = (Φ(, t)e i, e j ) L 2 and S i,j (t) = (Λ e i, e j ) (L 2 ) d, and F (t) = ((f(, t), e j )) j=1,...,nk. Since Φ ϕ and (e i ) i=1,...,nk is orthonormal for (, ) L 2, it holds M(t) ϕ Id. M(t) 1 is therefore well defined and measurable bounded over [, T ]. Hence, the initial value problem (6.6) can be put in standard form, with bounded measurable coefficients, and it therefore has a unique solution Θ such that Θ is bounded. There exists thus a unique solution ψ k to (6.59), with ψ k W 1, (, T ; V k ) W 1, (, T ; H 1 ()). Let us now prove some a priori estimates on ψ k. We

213 22 6 Degenerate parabolic problems make, for a.e. t (, T ), w = ψ k (t) in (6.59) and we integrate over t (τ, T ), for some τ (, T ). Since Λ is symmetric and does not depend on t, (Λ ψ k (t), ψ k(t)) (L2 ) = 1 d d 2 dt (Λ ψ k(t), ψ k (t)) (L2 ) d and we therefore obtain, using ψ k (, T ) = and the Young inequality (C.9), T τ Φ(x, t) t ψ k (x, t) 2 dxdt + 1 Λ(x) ψ k (x, τ) ψ k (x, τ)dx 2 f L2 ( (,T )) tψ k L2 ( (τ,t ) 1 f 2ϕ L 2 ( (,T )) + ϕ 2 tψ k 2 L 2 ( (τ,t ). This estimate holds for any τ (, T ). Given that Λ is uniformly coercive and that Φ ϕ, we deduce that (ψ k ) k N is bounded in L (, T ; H 1 ()) and that ( t ψ k ) k N is bounded in L 2 ( (, T )). Hence, there exists ψ L (, T ; H 1 ()) such that t ψ L 2 ( (, T )) and, up to a subsequence as k, ψ k ψ weakly- in L (, T ; H 1 ()) and t ψ k t ψ weakly in L 2 ( (, T )). Using Aubin Simon s theorem, we also see that the convergence of (ψ k ) k N holds in C([, T ]; L 2 ()), which ensures that ψ(, T ) =. We then take θ Cc (, T ) and v V l for some l N, and apply (6.59) for k l to θ(t)v instead of v. Integrating the resulting equation over t (, T ), we can take the limit and see that ψ satisfies, with ρ(x, t) = θ(t)v(x), T T Φ(x, t) t ψ(x, t)ρ(x, t)dxdt = T Λ(x) ψ(x, t) ρ(x, t)dxdt f(x, t)ρ(x, t)dxdt. (6.61) Any function ρ in L 2 (, T ; H 1 ()) can be approximated in this space by finite sums of functions (x, t) θ(t)v(x), with θ Cc (, T ) and v l N V l (see [31]). Hence, (6.61) also holds for any ρ L 2 (, T ; H 1 ()). Considering smooth compactly supported functions ρ, (6.61) shows that div(λ ψ) = f Φ t ψ in the sense of distributions. This proves that div(λ ψ) L 2 ( (, T )) and thus, by (6.61), that (6.58) is satisfied. Note that Lemma 6.29 below provides an additional regularity property and an integration-by-part formula on ψ. Step 2: proof of (6.57). Taking s, τ [, T ], we have τ w(x, t)div(λ ψ)(x, t)dxdt = s and τ s Λ(x) w(x, t) ψ(x, t)dxdt,

214 6.7 Proof of the uniqueness of the solution to the model 23 τ w(x, t) t ψ(x, t)dxdt = (w(x, τ)ψ(x, τ) w(x, s)ψ(x, s))dx s τ ψ(x, t) t w(x, t)dxdt. s Multiplying (6.56) by t ψ(x, t) + div(λ ψ)(x, t), integrating over (s, T ) for s [, T ], using (6.65) in Lemma 6.29, and recalling that ψ(, T ) =, we obtain 1 2 Λ(x) ψ(x, s) ψ(x, s)dx T + = s T s T s ( ( ) 2 ( ) ) 2 (1 g(x, t)) t ψ(x, t) + g(x, t) div(λ ψ)(x, t) dxdt Λ(x) w(x, t) ψ(x, t)dxdt w(x, s)ψ(x, s)dx ψ(x, t) t w(x, t)dxdt. (6.62) Integrating (6.62) with respect to s (, T ) leads to 1 T Λ(x) ψ(x, s) ψ(x, s)dxds 2 T T T Λ(x) w(x, t) ψ(x, t) dxdt + + T T w(x, s)ψ(x, s) dxds ψ(x, t) t w(x, t) dxdt. (6.63) Apply the Cauchy-Schwarz and Poincaré inequalities to obtain λ 2 ψ L 2 ( (,T )) T λ w d L 2 ( (,T )) + diam() ( w L 2 ( (,T )) + T tw L 2 ( (,T )) ). (6.64) Letting s = in (6.62), recalling that w(, ) =, and using (6.64) gives T ( ( ) 2 ( ) ) 2 (1 g(x, t)) t ψ(x, t) + g(x, t) div(λ ψ)(x, t) dxdt ( ) λ w L 2 ( (,T )) + diam() tw L 2 ( (,T )) ψ L 2 ( (,T )). d Combined with (6.64), this shows that (6.57) holds. Lemma Assume that, T and Λ satisfy (6.3a) and (6.3e). Let ψ L (, T ; H 1 ()) such that t ψ and div(λ ψ) belong to L 2 ( (, T )). Then ψ C([, T ]; H 1 ()) and, for all s, τ [, T ],

215 24 6 Degenerate parabolic problems τ t ψ(x, t)div(λ ψ)(x, t)dxdt = 1 Λ(x) ψ(x, τ) ψ(x, τ)dx s Λ(x) ψ(x, s) ψ(x, s)dx. (6.65) 2 Proof. Step 1: ψ C([, T ]; L 2 ()) and ψ : [, T ] H 1 () is continuous for the weak topology of H 1 (). Since ψ L (, T ; H 1 ()) L 2 (, T ; L 2 ()) and t ψ L 2 (, T ; L 2 ()), we have ψ H 1 (, T ; L 2 ()) C([, T ]; L 2 ()). Let M = ψ L (,T ;H 1()) and let t [, T ]. There exists (t n) n N converging to t such that ψ(t n ) H 1 () M. Since ψ is continuous with values in L2 (), we have ψ(t n ) ψ(t) in L 2 (). Given the bound on ψ(t n ) H 1 (), this convergence also holds in H 1 (), and ψ(t) H 1 () M. In other words, M is not just an essential bound of ψ( ) H 1 (), but actually a pointwise bound. Let us now prove the weak continuity of ψ. Let t [, T ] and t n t. If γ Cc () we have (ψ(t n ), γ) H 1 = ψ(x, t n ) γ(x)dxdt = ψ(x, t n ) γ(x)dxdt and thus, as n, since ψ C([, T ]; L 2 ()), (ψ(t n ), γ) H 1 ψ(x, t) γ(x)dxdt = ψ(x, t) γ(x)dxdt = (ψ(t), γ) H 1. (6.66) If γ H 1 () then we take γ ε Cc () such that γ γ ε H 1 () ε and we classically write (ψ(t n ), γ) H 1 (ψ(t), γ) H 1 (ψ(tn (ψ(t n ), γ) H 1 (ψ(t n ), γ ε ) H 1 + ), γ ε ) H 1 (ψ(t), γ ε ) H 1 + (ψ(t), γ ε ) H 1 (ψ(t), γ) H 1 Mε + (ψ(t n ), γ ε ) H 1 (ψ(t), γ ε ) H 1 + Mε. Taking the superior limit as n (using (6.66) with γ ε instead of γ), and then the limit as ε, we deduce that that (ψ(t n ), γ) H 1 (ψ(t), γ) H 1 as n. This concludes the proof of the continuity of ψ : [, T ] H 1 ()-w. Step 2: proof of (6.65). We only have to consider the case s < τ. We truncate ψ to [s, τ] and extend it by its constant values at the endpoints of this interval, which consists in defining ψ on R by

216 6.7 Proof of the uniqueness of the solution to the model 25 ψ(s) if t s, ψ(t) = ψ(t) if t (s, τ), ψ(τ) if t τ. Since ψ C([, T ]; L 2 ()) C([, T ]; H 1 ()-w), this definition makes sense and we have ψ C(R; L 2 ()) C(R; H 1 ()-w). By these continuity properties, we have t ψ = 1 (s,τ) t ψ since no Dirac masses are introduced at s or τ. We also have, on (s, τ), div(λ ψ) = div(λ ψ) L 2 ( (s, τ)). However, because we cannot ensure that div(λ ψ(τ)) and div(λ ψ(s)) belongs to L 2 (), we cannot say that div(λ ψ) L 2 ( R). We only have div(λ ψ) C(R; H 1 ()-w), owing to ψ C(R; H 1 ()-w). Let (ρ n ) n N be a smoothing kernel in time, such that supp(ρ n ) ( (τ s), ). We set ψ n (x, t) = (ψ(x, ) ρ n )(t). Then ψ n C (R; H 1 ()) and we can write, since Λ is symmetric and does not depend on time, τ s t ψ n (t), div(λ ψ n )(t) H 1,H 1dt τ = t ψ n (x, t) Λ(x) ψ n (x, t)dxdt s = 1 τ d Λ(x) ψ 2 s dt n (x, t) ψ n (x, t)dxdt = 1 Λ(x) ψ 2 n (x, τ) ψ n (x, τ)dx + 1 Λ(x) ψ 2 n (x, s) ψ n (x, s)dx. (6.67) We aim at passing to the limit n in this relation. By choice of supp(ρ n ) and by definition of ψ, ψ n (x, τ) = = R Hence, for all n N, 1 2 τ ψ(x, q)ρ n (τ q)dq ψ(x, q)ρ n (τ q)ds = ψ(x, τ) Λ(x) ψ n (x, τ) ψ n (x, τ)dx = 1 2 τ ρ n (τ q)dq = ψ(x, τ). Λ(x) ψ(x, τ) ψ(x, τ)dx. (6.68) Since ψ C(R; H 1 ()-w), as n we have ψ n (s) ψ(s) = ψ(s) weakly in H 1 (). The bilinear form (Λ, ) (L 2 ) being a Hilbert norm in d H1 (), we infer that

217 26 6 Degenerate parabolic problems 1 lim inf Λ(x) ψ n 2 n (x, s) ψ n (x, s)dx 1 Λ(x) ψ(x, s) ψ(x, s)dx. (6.69) 2 Dealing with the left-hand side of (6.67) is a bit more challenging, due to the lack of regularity of div(λ ψ) outside (s, τ). By definition of ψ, we have div(λ ψ(t)) = 1 (,s] (t)div(λ ψ(s)) + 1 (s,τ) (t)(div(λ ψ(t)) + 1 [τ,+ ) (t)(div(λ ψ(τ)). The choice of the support of ρ n ensures that, whenever t > s, 1 (,s] ρ n (t) =. Hence, for t (s, τ), div(λ ψ n (t)) = [ div(λ ψ( ))1 (s,τ) ] ρn (t) + (1 [τ,+ ) ρ n )(t)(div(λ ψ(τ)). Since div(λ ψ)1 (s,τ) L 2 ( R), the left-hand side of (6.67) can therefore be re-cast as τ s t ψ n (t), div(λ ψ n )(t) H 1,H 1dt τ = t ψ n (x, t) [ ] div(λ ψ)(x, )1 (s,τ) ρn (t)dxdt = s τ + s τ s t ψ n (t), div(λ ψ(τ)) H 1,H 1(1 [τ, ) ρ n )(t)dt t ψ n (x, t) [ ] div(λ ψ)(x, )1 (s,τ) ρn (t)dxdt + T n, (6.7) where T n = τ s F n(t)(1 [τ, ) ρ n )(t)dt with F n (t) = F ρ n (t), F (t) = ψ(t), div(λ ψ(τ) H 1,H 1. Integrating-by-parts, we have T n = F n (τ)(1 [τ, ) ρ n )(τ) F n (s)(1 [τ, ) ρ n )(s) τ s F n (t)(1 [τ, ) ρ n ) (t)dt. The choice of support of ρ n ensures that (1 [τ, ) ρ n )(s) = and that (1 [τ, ) ρ n )(τ) = 1. We also notice that (1 [τ, ) ρ n ) = δ τ ρ n has support in (s, τ) and converges weakly in the sense of measures toward the Dirac mass δ τ. Since ψ C(R; H 1 ()-w), we have F C(R) and thus F n F locally uniformly on R. Hence, as n, T n = F n (τ) τ s F n (t)(1 [τ, ) ρ n ) (t)dt F (τ) F (τ) =. (6.71)

218 6.8 Numerical example 27 The functions t ψ and div(λ ψ)(x, )1 (s,τ) belong to L 2 ( R), so t ψ n = ( t ψ) ρ n t ψ in L 2 ( R) and [ ] div(λ ψ)(x, )1(s,τ) ρn div(λ ψ)(x, )1 (s,τ) in L 2 ( R). Using (6.71), we can therefore pass to the limit in (6.7) and we see, since t ψ = t ψ on (s, τ), τ lim n s t ψ n (t), div(λ ψ n )(t) H 1,H 1dt τ = t ψ(x, t)div(λ ψ)(x, t)dxdt. Combined with (6.67), (6.68) and (6.69), this gives (6.65) with instead of =. The converse inequality is obtained by re-doing the previous reasoning with smoothing kernels ρ n having support in (, τ s), or by reversing the time as at the end of the proof of Lemma Step 3: proof that ψ : [, T ] H 1 () is continuous for the strong topology of H 1 (). Since the left-hand side of (6.65) is continuous with respect to s, the mapping s (Λ ψ(s), ψ(s)) (L2 ) d is continuous. Assume that s n s in [, T ]. Owing to ψ C([, T ]; H 1 ()-w) we have ψ(s n ) ψ(s) weakly in H 1 (). Moreover, (Λ ψ(s n ), ψ(s n )) (L 2 ) d (Λ ψ(s), ψ(s)) (L 2 ) d. Since (Λ, ) (L 2 ) d is a Hilbert norm on H 1 (), we conclude that ψ(s n ) ψ(s) strongly in H 1 (). s 6.8 Numerical example We consider a 2D test case with β(s) = s and Λ = I d, which means that we approximate the Stefan problem. The scheme used here is the VAG scheme described in Section 8.5. The domain is = (, 1) 2, and we use the following definition of ζ(u), u if u <, ζ(u) = u 1 if u > 1, otherwise. Dirichlet boundary conditions are given by u = 1 on and the initial condition is u(x, ) = 2. Four grids are used for the computations: a Cartesian grid with 32 2 = 124 cells, the same grid randomly perturbed, a triangular grids with 896 cells, and a Kershaw mesh with 189 cells as illustrated in Figure 6.4 (such meshes are standard in the framework of underground

219 28 6 Degenerate parabolic problems engineering). The time simulation is.1 for a constant given time step of.1. Figures 6.4, 6.5, 6.6 and 6.7 represent the discrete solution u(, t) on all grids for t =.25,.5,.75 and.1. For a better comparison we have also plotted the interpolation of u along two lines of the mesh. The first line is horizontal and joins the two points (,.5) and (1,.5). The second line is diagonal and joins points (, ) and (1, 1). The results for these slices are shown in Figures 6.2 and 6.3. The numerical outputs are weakly dependent on the grid, and the interface between the regions u < and u > 1 are located at the same place for all grids. It is worth noticing that this remains true even for the very irregular Kershaw mesh (which presents high regularity factors κ T see (7.1), that is high ratios for some cells between the radii of inscribed balls and the diameter of the cell). (a) t =.25 (b) t =.5 (c) t =.75 (d) t =.1 Fig Interpolation of u along the line x 2 =.5 of the mesh for each grids : Cartesian in blue, perturbed Cartesian in red, triangular in green, and Kershaw in black dashed.

220 6.8 Numerical example 29 (a) t =.25 (b) t =.5 (c) t =.75 (d) t =.1 Fig Interpolation of u along a diagonal axe of the mesh for each grids: Cartesian in blue, perturbed Cartesien in red, triangular in green, and Kershaw in black dashed.

221 21 6 Degenerate parabolic problems (a) Cartesian (b) Perturbed Cartesian (c) Triangular (d) Kershaw Fig Discrete solution u on all grids at t =.25. (a) Cartesian (b) Perturbed Cartesian (c) Triangular (d) Kershaw Fig Discrete solution u on all grids at t =.5.