Finite Volume Methods Schemes and Analysis course at the University of Wroclaw

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1 Finite Volume Methods Schemes and Analysis course at the University of Wroclaw Robert Eymard 1, Thierry Gallouët 2 and Raphaèle Herbin 3 April, Université Paris-Est Marne-la-Vallée 2 Université de Provence 3 Université de Provence

2 Contents Introduction 3 1 Finite volume discretization of balance laws A generic example Finite volume meshes and space-time discretizations Linear diffusion problems Actual problems and schemes Numerical results Further analysis The convergence proof in the (harmonic averaging) isotropic case The convergence proof in the case of the gradient schemes The convergence proof in the case of the mixed-type schemes Nonlinear convection - degenerate diffusion problems Actual problems and schemes Finite volume approximation A numerical example Further analysis Continuous definitions and main convergence result Existence, uniqueness and discrete properties Compactness of a family of approximate solutions Convergence towards an entropy process solution Uniqueness of the entropy process solution Conclusion Navier-Stokes equations Actual problems and schemes Finite volume scheme with staggered variables The MAC scheme on regular grids An example, built on the pressure grid Further analysis Finite volume scheme with collocated variables Discrete scheme Further analysis Numerical example

3 2 5 Discrete functional analysis The topological degree argument Discrete Sobolev embedding Discrete embedding of W 1,1 () in L 1 () Discrete embedding of W 1,p () in L p (), 1 < p < d Discrete embedding of W 1,p () in L q (), for some q > p Compactness results for bounded families in discrete W 1,p () norm Compactness in L p () Regularity of the limit Properties in the case of the adapted discretizations Lemma used for time translates Bibliography 83

4 Introduction Our aim is to describe and analyse numerical schemes, issued from cell centred finite volume methods, applied to various physical problems. For each problem, we present a mathematical model and some relevant finite volume schemes. The efficiency of the schemes is then illustrated by some numerical results. In a second step, a mathematical analysis of the schemes is carried out. The first chapter introduces the notion of mesh, suitable for finite volume discretizations. In the second and third chapters, we focus on problems which are modeled by a scalar partial differential equation: we shall successively deal with pure diffusion operators, isotropic or not, and with convection-diffusion operators including the pure hyperbolic case. In the fourth chapter, we focus on the incompressible viscous flow (Navier-Stokes equations). We finally provide, in the fifth chapter, some mathematical results, used in the convergence and error analysis of the schemes. 3

5 Chapter 1 Finite volume discretization of balance laws 1.1 A generic example Let us take a generic example, dedicated to show the need of complete definitions for the space and time discretizations. Let be a polygonal open subset of R d, T R, and let us consider a scalar balance law written under the general form: u t + div(f (u, u)) + s(u) = 0 on (0, T ), (1.1) where F C 1 (R R d, R d ) and s C(R, R), and u is a real function of (x, t) (0, T ). In order to give the main ideas of the finite volume schemes, we need to introduce the set M containing the control volumes, which are the subsets of on which balance equations are written (see Definition 1.1). Such a balance equation is obtained from the above conservation law by integrating it over a control volume M and applying the Stokes formula: u t dx + F (u, u) n dγ(x) + s(u) dx = 0, where n stands for the unit normal vector to the boundary outward to and γ denotes the integration with respect to the (d 1) dimensional Lebesgue measure. Let us denote by E the set of edges (faces in 3D) of the mesh, and E the set of edges which form the boundary of the control volume. With these notations, the above equation reads: u t dx + F (u, u) n dγ(x) + s(u) dx = 0. σ σ E Let δt = T/M, where M N, M 1, and let us perform an explicit Euler discretization of the above equation (an implicit or semi-implicit discretization could also be performed, and is sometimes preferable, depending on the type of equation). We then get: u (n+1) u (n) dx + F (u (n), u (n) ) n dγ(x) + s(u (n) ) dx = 0, δt σ σ E where u (n) denotes an approximation of u(, t (n) ), with t (n) = nδt (see Definition 1.6 for a general definition of a time discretization). Let us then introduce the discrete unknowns (one per control volume and time step) (u (n) ) M, n N; assuming the existence of such a set of real values, we may define a piecewise constant function by: u (n) M H M() : u (n) M = 4 M u (n) 1,

6 5 where H M () denotes the space of functions from to R which are constant on each control volume of the mesh M. In order to define the scheme, the fluxes σ F (u(n), u (n) ) n dγ(x) need to be approximated as a function of the discrete unknowns. We denote by F,σ (u (n) M ) the resulting numerical flux, the expression of which depends on the type of flux to be approximated. Let us now give this expression for various simple examples. First we consider the case of a linear convection equation, that is equation (1.1) where the flux F (u, u) reduces to F (u, u) = vu, v R d, and s(u) = 0: u t + div(vu) = 0 on. (1.2) In order to approximate the flux vu n on the edges of the mesh, one needs to approximate the value of u on these edges, as a function of the discrete unknowns u associated to each control volume. This may be done in several ways. A straightforward choice is to approximate the value of u on the edge σ = L separating the control volumes and L by the mean value 1 2 (u + u L ). This yields the following numerical flux: F (cv,c),σ (u u + u L M) = v,σ 2 where v,σ = σ v n,σ, and n,σ denotes the unit normal vector to the edge σ outward to. This centred choice is known to lead to stability problems, and is therefore often replaced by the so called upstream choice, which is given by: F (cv,u),σ (u M ) = v +,σ u v,σ u L, (1.3) where x + = max(x, 0) and x = min(x, 0). Hence we see that for purely convective problems, we do not require many geometrical properties on the mesh. Such meshes are formally defined in definition 1.1, where we only need to give sufficient properties in order to compute the normal to the boundary. If we now consider a linear convection diffusion reaction equation, that is equation (1.1) with F (u, u) = Λ u + vu, v R d, and s(u) = bu, b R: σ u t + div( Λ u + vu) + bu = 0 on, (1.4) the flux through a given edge then reads: F (u) n,σ = ( Λ u + v u) n,σ, σ so that we now need to discretize the additional term σ Λ u n,σ; this diffusion flux involves the knowledge of the derivatives at the boundary. In order to give approximations of this term, we need to define finite differences, consistent with u, which implies that u is evaluated at some particular points of the mesh. This is the purpose of Definition of pointed meshes (see Definition 1.2 and the following ones, each of them suited with particular meshes frameworks). 1.2 Finite volume meshes and space-time discretizations As announced in the first section of this chapter, we now present the precise definitions of space and time discretizations, used all along this book for defining the different schemes. General finite volume meshes are used for partial differential equations including first order, second order (scarcely more). In the case of convection problems, we only need of a partition of the domain in control volumes, whereas in the case of diffusion problems, we also need to define points into the control volumes (see below). We shall consider a series of properties, and a discretisation will be defined by reference to some of them. Let us consider that d N, R d is open, polygonal, bounded, connected, = \ is the boundary of, assumed to be Lipschitz-continuous. (1.5)

7 6 In the above definition, polygonal means that is a finite union of subsets of hyperplanes of R d. Let us give a definition for a polygonal finite volume discretisation of. Definition 1.1 (Polygonal finite volume space discretization of ) Under hypothesis (1.5) a polygonal finite volume discretization of, is given by the pair (M, E), where: 1. M is a finite family of non empty connected open disjoint subsets of (the control volumes ) such that = M. For any M, let = \ be the boundary of, m() > 0 denote the measure of and h denote the diameter of. 2. E is a finite family of disjoint subsets of (the edges of the mesh), such that, for all σ E, σ is a non empty open subset of a hyperplane of R d, whose (d-1)-dimensional measure m(σ) is stricly positive. We assume that, for all M, there exists a subset E of E such that = σ E σ. We then denote by M σ = { M, σ E }. We then assume that, for all σ E, either M σ has exactly one element and then σ (the set of these edges, called boundary edges, is denoted by E ext ) or M σ has exactly two elements (the set of these edges, called interior edges, is denoted by E int ). For all M and σ E, we denote for a.e. x σ by n,σ the unit vector normal to σ outward to. For all M, we denote by N the set of the neighbours of : N = {L M \ {}, σ E int, M σ = {, L}}. (1.6) We then denote by H M () the set of functions from to R, constant on each element of M. The size of the discretization is defined by: h(m) = sup{diam(), M}. (1.7) We then define the space X D = R M E, containing all the families of reals ((u ) M, (u σ ) σ E ), and we consider the subspace X D,0 of all families ((u ) M, (u σ ) σ E ) such that u σ = 0 if σ E ext. We then define, for u X D, Π M u H M () as the element of H M () equal a.e. in to the value u, for all M. In order to study the schemes introduced in this paper for diffusion operators, we also define the notion of pointed finite volume discretization. Definition 1.2 (Pointed polygonal finite volume space discretization of ) Under hypothesis (1.5), a pointed polygonal finite volume discretization of, is given by the triplet (M, E, P), where: 1. (M, E) is a polygonal finite volume discretization of in the sense of definition P is a family of points of indexed by M and E, denoted by P = ((x ) M, (x σ ) σ E ), such that for all M, x and for all σ E, x σ σ. We then denote by d,σ the abscissae of x on the straight line with origine x σ and oriented by n,σ. We then define P M : C() H M () by P M ϕ(x) = ϕ(x ) for a.e. x, M, ϕ C(). (1.8) and P D : C() X D by P D ϕ = ϕ(x ), M, P D ϕ σ = ϕ(x σ ), σ E, ϕ C(). (1.9) Remark 1.1 The conditions x and x σ σ can easily be relaxed.

8 7 x d σ xσ Figure 1.1: Pointed mesh Thanks to the algebraic definition of d,σ, we have the relation σ E m(σ)d,σ = d m(), M. (1.10) Definition 1.3 (Pointed star-shaped polygonal finite volume space discretization of ) Under hypothesis (1.5), a pointed polygonal finite volume discretization of, denoted by (M, E, P), is said to be star-shaped, if all M is x -star-shaped, which means that for all x, the property [x, x] holds (this is equivalent to d,σ 0 for all σ E ). A pointed polygonal finite volume discretization of, denoted by (M, E, P), is said to be strictly star-shaped if, for all M and σ E, d,σ > 0. For all M and σ E, we denote by D,σ the cone with vertex x and basis σ D,σ = {tx + (1 t)y, t (0, 1), y σ}, (1.11) and we then define the set L D () of all functions g which are constant in each cone D,σ (we then denote by g,σ the corresponding value). Note that H M () L D (). We denote, for all σ E, D σ = M σ D,σ (this set is sometimes called the diamond associated to the edge σ). We then define d σ, for all σ E, by d σ = d L,σ + d,σ, d σ = d,σ, σ E int, M σ = {, L}, σ E ext, M σ = {}. (1.12) Remark 1.2 The above definition applies to a large variety of meshes. Note that no hypothesis is made on the convexity of the control volumes, which enables that hexahedra with non planar faces can be used (in fact, such sets have then 12 faces if each non planar face is shared in two triangles, but only 6 neighbouring control volumes). Remark 1.3 The common boundary of two neighbouring control volumes can include more than one edge. Definition 1.4 ( adapted pointed polygonal finite volume space discretization of ) Under hypothesis (1.5), a strictly star-shaped pointed polygonal finite volume discretization of, denoted by (M, E, P), is said to be a adapted pointed polygonal finite volume space discretization of, if, for all σ E int, denoting by, L M the two control volumes such that M σ = {, L}, then the straight line (x, x L ) is orthogonal to σ. In this case, we denote by x σ the intersection between σ and the line (x, x L ) (assumed to belong to σ for the sake of simplicity). Then, for, L M, such that there exists σ E int with M σ = {, L}, σ is unique and one denotes σ = L. For all M and σ E E ext, we denote by x σ the orthogonal projection of x on σ. An example of two neighbouring control volumes and L of M is depicted in Figure 1.2.

9 8 m(σ) x L x L d L L d,σ Figure 1.2: Notations for a -adapted mesh Definition 1.5 ( super-adapted pointed polygonal finite volume space discretization of ) Under hypothesis (1.5), a adapted pointed polygonal finite volume discretization of, denoted by (M, E, P), is said to be super adapted, if, for all σ E int, denoting by, L M the two control volumes such that M σ = {, L}, then the straight line (x, x L ) is orthogonal to σ and intersects σ at the point x σ, equal to the center of gravity of σ. Definition 1.6 (Time discretization of (0, T )) A time discretization of (0, T ) is given by an integer value N and by an increasing sequence of real values (t n ) n [0,N+1 ] with t 0 = 0 and t N+1 = T. The time steps are then defined by δt n = t n+1 t n, for n [0, N]]. Definition 1.7 (Space-time discretization of (0, T )) A finite volume discretization D of (0, T ) is the family D = (M, E, (x ) M, N, (t n ) n [0,N ] ), where M, E, (x ) M is a finite volume mesh of in the sense of one of the Definitions and N, (t n ) n [0,N+1 ] is a time discretization of (0, T ) in the sense of Definition 1.6. For a given mesh D, one defines: D = max(h(m), (δt n ) n [0,N ] )

10 Chapter 2 Linear diffusion problems 2.1 Actual problems and schemes We wish to develop and study some schemes for the approximation of the weak solution ū of the following diffusion problem with full anisotropic tensor: div(λ ū) = f in, (2.1) ū = 0 on, under the following assumptions: is an open bounded connected polygonal subset of R d, d N, (2.2) and Λ is a measurable function from to M d (R), where M d (R) denotes the set of d d matrices, such that for a.e. x, Λ(x) is symmetric, the lowest and the largest eigenvalues of Λ(x), denoted by λ(x) and λ(x), are such that λ, λ L () and there exists λ 0 R with 0 < λ 0 λ(x) λ(x) for a.e. x, We give the classical weak formulation in the following definition. (2.3) f L 2 (). (2.4) Definition 2.1 (Weak solution) Under hypotheses (2.2)-(2.4), we say that ū is a weak solution of (2.1) if ū H0 1 (), Λ(x) ū(x) v(x)dx = f(x)v(x)dx, v H0 1 (). (2.5) Remark 2.1 For the sake of clarity, we restrict ourselves here to the numerical analysis of Problem (2.1), however, the present analysis readily extends to convection-diffusion-reaction problems and coupled problems. Indeed, we emphasize that proofs of convergence or error estimate can easily be adapted to such situations, since the discretization methods of all these terms are independent of one another, and the treatment of convection and reaction term is well-known exact (see [26] or [20]). We consider in this chapter a family of schemes based on the definition of a numerical flux, using an unknown located at the edges of the mesh. A variety of situations with respect to a possible elimination of this unknown are then considered, within always ensuring the following properties: 1. The matrices of the generated linear systems are expected to be sparse, symmetric, positive and definite. 9

11 10 2. We wish to be able to prove the convergence of the discrete solution and an associate gradient to the solution of the continuous problem and its gradient, and to show error estimates. Under hypothesis (1.5), let D = (M, E, P) be a discretization of in the sense of definition 1.2 (we shall also consider below the particular case of adapted discretizations in the sense of definition 1.4). Common features of the schemes The idea of the schemes described in this chapter is to find an approximation of the solution of (2.1) by setting up a system of discrete equations for a family of values ((u ) M, (u σ ) σ E ) in the control volumes and on the interfaces. Without a suitable elimination procedure, the number of unknowns is therefore card(m) + card(e). Following the idea of the finite volume framework, equation (2.1) is integrated over each control volume M, which formally gives (assuming sufficient regularity on u and Λ) the following balance equation on the control volume : ( ) Λ(x) u(x) n,σ dγ(x) = f(x)dx. σ σ E The flux σ Λ(x) u(x) n,σdγ(x) is approximated by a function F,σ (u) of the values ((u ) M, (u σ ) σ E ) at the centers and at the interfaces of the control volumes (in all the cases presented below, F,σ (u) only depends on u and all (u σ ) σ E ). We give the définition of the sets X D and X D,0 (also provided in Definition (1.1)) where the discrete unknowns lie, that is to say: X D = {v = ((v ) M, (v σ ) σ E ), v R, v σ R}, (2.6) X D,0 = {v X D such that v σ = 0 σ E ext }. (2.7) A discrete equation corresponding to (2.1) is then: find u X D,0 such that F,σ (u) = f(x)dx, M. (2.8) σ E The values u σ on the interfaces are then introduced so as to allow for a consistent approximation of the normal fluxes in the case of an anisotropic operator and a general, possibly nonconforming mesh. Thanks to the definition of X D,0, the values u σ for the boundary faces or edges are prescribed by the discrete counterpart of the homogeneous Dirichlet boundary condition: u σ = 0, σ E ext. (2.9) We then need card(e int ) equations to ensure that the problem is well posed. Following the finite volume ideas, we may write the continuity of the discrete flux for all interior edges, that is to say: F,σ (u) + F L,σ (u) = 0, for σ E int such that M σ = {, L}. (2.10) We now have card(m) + card(e int ) unknowns and equations. Note that u X D,0 is solution of (2.8)-(2.9)-(2.10) if and only if it satisfies u X D,0, u, v F = v f(x)dx, for all v X D,0, (2.11) where we set M u, v F = M σ E F,σ (u)(v v σ ). (2.12) Indeed, taking v = 1, v L = 0 for all L M with L and v σ = 0 for all σ E leads to (2.8). Taking v σ = 1 for σ E int, v σ = 0 for all σ E with σ σ and v = 0 for all M provides (2.10). Reciprocally, for all v X D,0, multiplying (2.8) by v and using (2.10) gives (2.11). In the following, we review different choices for the meshes and for the expression of the numerical fluxes, leading to different properties for the resulting scheme. Let us first turn to the simple case of isotropic diffusion, using adapted pointed meshes.

12 11 The case of the Laplace operator on adapted pointed meshes We assume in this section the hypothesis (2.2) and that Λ(x) = Id where Id is the identity application of R d for a.e. x. (2.13) Let D be an admissible discretization of in the sense of Definition 1.4. The finite volume approximation to Problem (2.1) is given as the solution u = ((u ) M, (u σ ) σ E ) X D,0 of (2.8)-(2.9)-(2.10), in which we define the numerical flux by: F,σ (u) = m(σ) u σ u d,σ, M, σ E. (2.14) The mathematical analysis of this scheme is given in section Nevertheless, let us rewrite the scheme, remarking that we can immediately eliminate u σ for M σ = {, L} from the relation (2.10). Indeed we get u σ = d L,σu + d,σ u L d L,σ + d,σ. (2.15) Hence we can identify the set of all elements of X D,0, such that the condition (2.15) is ensured, with the set H M (), defined in Definition 1.1 as the set of the piecewise constant functions in each control volume. Reporting the above expression of u σ given by (2.15) in (2.14), we get where we define, for all u H M () and where we recall that F,σ (u) = m(σ) δ,σu d σ, M, σ E, (2.16) δ,σ u = u L u, δ,σ u = u, d σ = d L,σ + d,σ, d σ = d,σ, σ E int, M σ = {, L}, σ E ext, M σ = {}, σ E int, M σ = {, L}, σ E ext, M σ = {}. (2.17) Then the solution of (2.8) is also solution of the classical 2-point flux finite volume scheme studied for example in [20]: find u H M (), such that m(σ) δ,σu = f(x)dx, M. d σ σ E Recall that the above scheme also writes u H M (), M δ,σu m(σ)d,σ d σ σ E The case of harmonic averaging of an isotropic tensor We assume in this section the hypothesis (2.2) and that δ,σ v = f(x)v(x)dx, v H M (). (2.18) d σ Λ(x) = λ(x)id where Id is the identity application of R d with λ L () such that there exists λ 0 (0, + ) with λ 0 λ(x) for a.e. x. (2.19) We consider in this section the so-called harmonic averaging case. Let D be an admissible adapted discretization of in the sense of Definition 1.4. The finite volume approximation to Problem (2.1) is given as the solution u = ((u ) M, (u σ ) σ E ) X D,0 of (2.8)-(2.9)-(2.10), in which we define the numerical flux by: F,σ (u) = m(σ)λ u σ u d,σ, M, σ E. (2.20)

13 12 In (2.20), we set λ = 1 λ(x)dx, M. m() Note that, eliminating u σ for all σ E int leads to find u H M () such that λ σ m(σ) δ,σu = f(x)dx, M, (2.21) d σ σ E denoting by λ σ, for all σ E, the harmonic averaged value defined by: λ d,σ Then the solution of (2.21) is also solution of u D H M (), M λ L d L,σ λ σ = d σ, σ E int, M σ = {, L} λ d,σ + λ L d L,σ λ σ = λ, σ E ext, M σ = {}. δ,σu λ σ m(σ)d,σ d σ σ E Elimination of u σ in the general case (2.22) δ,σ v = f(x)v(x)dx, v H M (). (2.23) d σ In the case where the relation (2.19) does not hold, we are then led to look for an expression of F,σ (u) depending on u and all (u σ ) σ E. In this case, it is no longer possible to locally eliminate u σ for all σ E int, only using (2.10). Remark 2.2 Note that in the case of regular simplicial conforming meshes (triangles in 2D, tetrahedra in 3D), there is an algebraic possibility to express the unknowns (u σ ) σ E as local affine combinations of the values (u ) M [35] and therefore to eliminate them. The idea is to remark that the linear system constituted by the equations (2.8) for all M S, where M S is the set of all simplices sharing the same interior vertex S, and (2.10) for all the interior edges such that M σ M S, presents as many equations as unknowns u σ, for σ MS E. Indeed, the number of edges in MS E such that M σ M S is equal to the number of control volumes in M S. Unfortunately, there is at this time no general result on the invertibility nor the symmetry of the matrix of this system, and this method does not apply to other types of meshes than the simplicial ones. We may then choose to use the weak discrete form (2.12) as an approximation of the bilinear form a(, ), but with a space of dimension smaller than that of X D,0. This can be achieved by expressing the value of u on any interior interface σ E int as a consistent barycentric combination of the values u : u σ = βσ u, (2.24) M where (βσ ) M is a family of real numbers, with βσ 0 only for some control volumes close to σ, and such σ E int that = 1 and x σ = βσ x, σ E int. (2.25) M β σ We recall that the values u σ, σ E ext are set to 0 in order to respect the boundary conditions. This ensures that if ϕ is a regular function, then ϕ σ = M β σ ϕ(x ) is a consistent approximation of ϕ(x σ ) for σ E int. Hence the new scheme reads: Find u X D,0 such that u σ = βσ u, σ E int, and M u, v F = v f(x)dx, for all v X D,0 with v σ = (2.26) βσ v, σ E int. M M M

14 13 This method has been shown in [25] to be efficient in the case of a problem where Λ = Id (for the approximation of the viscous terms in the Navier-Stokes problem). Unfortunately, because of a poor approximation of the local flux at strongly hetereogenous interfaces, this approach is not sufficient to provide accurate results for some types of flows in heterogeneous media, as we shall show in section 2.2. This is especially true when using coarse meshes, as is often the case in industrial problems. Therefore it is possible to take advantage of both techniques: we shall use equation (2.12) and keep the unknowns u σ on the edges which require them, for instance those where the matrix Λ is discontinuous: hence (2.10) will hold for all edges associated to these unknowns; for all other interfaces, we shall impose the values of u using (2.24), and therefore eliminate these unknowns. Let us decompose the set E int of interfaces into two non intersecting subsets, that is: E int = A E hyb, E hyb = E int \ A. The interface unknowns associated with A will be computed by using the barycentric formula (2.24). Remark 2.3 Note that, although the accuracy of the scheme is increased in practice when the points where the matrix Λ is discontinuous are located within the set σ E hyb σ, such a property is not needed in the mathematical study of the scheme. Let us introduce the space X D,A X D,0 defined by: X D,A = {v X D such that v σ = 0 for all σ E ext and v σ satisfying (2.24) for all σ A}. (2.27) The composite scheme which we consider in this work reads: { Find u XD,A such that: u, v F = M v f(x)dx, for all v X D,A. (2.28) We therefore obtain a scheme with card(m) + card(e hyb ) equations and unknowns. It is thus less expensive while it remains precise (for the choice of numerical flux given below) even in the case of strong heterogeneities (see section 2.2). Note that with the present scheme, (2.10) holds for all σ E hyb, but not generally for any σ A. However, fluxes between pairs of control volumes can nevertheless be identified. These pairs are no longer necessarily connected by a common boundary, but determined by the stencil used in relation (2.24). Remark 2.4 (Other boundary conditions) In the case of Neumann or Robin boundary conditions, the discrete space X D,A is modified to include the unknowns associated to the corresponding edges, and the resulting discrete weak formulation is then straightforward. Remark 2.5 (Extension of the scheme) There is no additional difficulty to replace (2.24) in the definition of (2.27) by u σ = βσ u + σ u σ, σ A, with M β σ + β σ σ σ E hyb M σ E hyb β σ = 1 and x σ = M Then all the mathematical properties shown below still hold. β σ x + β σ σ E hyb σ x σ, σ A. Let us apply these principles in two situations. The first one is the case of an anisotropic tensor and of a adapted mesh, for which we shall be able to get back a standard finite volume scheme. The second one is that of an anisotropic tensor and of a general pointed mesh. The case of an anisotropic tensor and of a adapted mesh Under hypotheses (2.2)-(2.4), let D be a adapted finite volume discretization of in the sense of Definition 1.4. For the sake of simplicity, we set A = E int, consider the space X D,Eint X D,0 defined by: X D,Eint = {v X D such that v σ = 0 for all σ E ext and v σ = d,σv L + d L,σ v d,σ + d L,σ for all σ E int }. (2.29)

15 14 Indeed, it is possible, in the particular case of a adapted finite volume discretization of, to use the property that x σ is located, for M σ = {, L}, on the line (x, x L ), and therefore, express (2.24) using only the two values u and u L. We then introduce a discrete gradient D : X D,Eint H M () d, given by constant values in each control volume: u = 1 m(σ)(x σ x ) u σ u, M, u X D,Eint. (2.30) m() d,σ σ E A first natural scheme, inspired by the finite element framework, would be u X D,Eint, Λ(x) D u(x) D v(x)dx = f(x)π M v(x)dx, v X D,Eint. (recall that Π M v is defined in Definition 1.1). But this scheme is not convenient, since it is possible to get D u = 0 with u 0. Hence a stabilization is necessary. To this purpose, we define an element R D u L D (), given by the expression R,σ u = ( ) uσ u α d u n,σ, M, σ E, u H M (). (2.31) d,σ where (α ) M is a family of given positive real values. The modified finite volume approximation to Problem (2.1) is given, for a given family of positive reals (α ) M, as the solution of the following equation: u X D,Eint, (2.32) (Λ(x) D u(x) D v(x) + R D u(x)r D v(x)) dx = f(x)π M v(x)dx, v X D,Eint. Remark 2.6 In the case Λ = Id, if D is a superadapted discretization in the sense of definition 1.5 (recall that it means x σ x = d,σ n,σ, for all M and σ E ) and α = 1 for all M, then the scheme (2.32) gives back the scheme provided above for the Laplace case. Indeed, we get that ( ) ( uσ u vσ v σ E m(d,σ )R,σ ur,σ v = which leads to m(σ)d,σ d σ E σ E m(d,σ )R,σ ur,σ v = d d,σ u σ u m(σ)d,σ d,σ σ E u n,σ d,σ v n,σ v σ v d,σ m() u v, using u σ E m(σ)d σ u,σ d,σ n,σ = m() u and σ E m(σ)d,σ n,σ n t,σ = m()id. In the sequel, we shall refer to the scheme (2.32) as the gradient scheme on adapted meshes, since it is obtained from a discretization formula on the gradient, the consistence of which results from the orthogonality property of the mesh. We may note that scheme (2.32) leads to ( m()λ u v setting M +α σ E m(σ)d,σ (u σ u d,σ u n,σ )(v σ v d,σ Λ = 1 Λ(x)dx. m() v n,σ ) ) = M v f(x)dx, ),

16 15 This expression of the scheme can then be put under the form σ (u σ u )(v σ v ) = v A σ M σ E σ E M f(x)dx, where the matrix (A σ σ ) σ,σ E is symmetric. Let us define F,σ (u) = σ (u σ u ), σ E, M. (2.33) σ E A σ Taking for v X D,Eint, v = 1 and v L = 0 for all L M \ {}, we obtain that the above equation is equivalent to finding the values (u ) M, solution of the following system of equations: F,σ (u) = f(x)dx, M, (2.34) σ E where and F,σ (u) = d,σ d L,σ F,σ (u) F L,σ (u), σ E int, M σ = {, L} (2.35) d,σ + d L,σ d,σ + d L,σ F,σ (u) = F,σ (u), σ E ext, M σ = {}. (2.36) This is indeed a finite volume scheme (in the sense of the respect of the local balance), since F,σ (u) = F L,σ (u), σ E int, M σ = {, L}. Gradient schemes on general non matching grids We now finally relax the assumption that it was possible to mesh the domain using a adapted mesh: indeed, actual problems involve more and more often general grids, which do not satisfy this orthogonality condition. This is for instance the case in subsurface flow modelling, where parallelepipedic (or quadrangular in 2D) meshes are widely used to mesh the underground layers. In the case of geologigal faults for instance, the independent meshing of geological layers results in non conforming meshes (see e.g. Figure 2.1) which are clearly not adapted meshes. Figure 2.1: Non conforming subsurface flow modelling Hence we consider that D = (M, E, P) is only a discretization of in the sense of definition 1.2, and we then denote by x σ the center of gravity of σ E. As we did in the case of an anisotropic diffusion, on adapted meshes, we again identify the numerical fluxes F,σ (u) through the mesh dependent bilinear form, F defined in (2.12), using the expression of a discrete gradient. Indeed let us assume that, for all u X D, we have constructed a discrete gradient D u, we then seek a family (F,σ (u)) M such that M σ E F,σ (u)(v v σ ) = D u(x) Λ(x) D v(x)dx, u, v X D. (2.37) σ E

17 16 Remark 2.7 It is then always possible to deduce an expression for F,σ (u) satisfying (2.37), under the sufficient condition that, for all M and a.e. x, D u(x) may be expressed as a linear combination of (u σ u ) σ E, the coefficients of which are measurable bounded functions of x. This property is ensured in the construction of D u(x) given below. Then, in order to ensure the desired properties 2 and 3, we shall see in Section that it suffices that the discrete gradient satisfies the following properties. 1. For a sequence of space discretisations of with mesh size tending to 0, if the sequence of associated grid functions is bounded in some sense, then their discrete gradient is expected to converge, at least weakly in L 2 () d, to the gradient of an element of H 1 0 (); 2. If ϕ is a regular function from to R, the discrete gradient of the piecewise function defined by taking the value ϕ(x ) on each control volume and ϕ(x σ ) on each edge σ is a consistent approximation of the gradient of ϕ. Le us first define: u = 1 m() σ E m(σ)(u σ u )n,σ, M, u X D, (2.38) where n,σ is the outward to normal unit vector, m() and m(σ) are the usual measures (volumes, areas, or lengths) of and σ. The consistency of formula (2.38) stems from the following geometrical relation: σ E m(σ)n,σ (x σ x ) t = m()id, M, (2.39) where (x σ x ) t is the transpose of x σ x R d, and Id is the d d identity matrix. Indeed, for any linear function defined on by ψ(x) = G x with G R d, assuming that u σ = ψ(x σ ) and u = ψ(x ), we get u σ u = (x σ x ) t G = (x σ x ) t ψ, hence (2.38) leads to u = ψ. Since the coefficient of u in (2.38) is in fact equal to zero, a reconstruction of the discrete gradient D u solely based on (2.38) cannot lead to a definite discrete bilinear form in the general case. Hence, we now introduce: with R,σ u =,σ u = u + R,σ u n,σ, (2.40) d d,σ (u σ u u (x σ x )), (2.41) (recall that d is the space dimension and d,σ is the Euclidean distance between x and σ). We may then define D u as the piecewise constant function equal to,σ u a.e. in the cone D,σ with vertex x and basis σ: D u(x) =,σ u for a.e. x D,σ. (2.42) We can then prove that the discrete gradient defined by (2.38)-(2.42) meets the required properties (see Lemmas 2.6 and 2.7). In order to identify the numerical fluxes F,σ (u) using the relation (2.37), we put the discrete gradient under the form,σ u = (u σ u )y σσ, σ E with y σσ = ( m(σ) d m() n,σ + 1 m(σ) d,σ m(σ ) m() n,σ Thus: D u(x) Λ(x) D v(x)dx = ) m() n,σ (x σ x ) n,σ if σ = σ d d,σ m() m(σ )n,σ (x σ x )n,σ otherwise. A σσ M σ E σ E (2.43) (u σ u )(v σ v ), u, v X D, (2.44)

18 17 with: A σσ = y σ σ Λ,σ y σ σ and Λ,σ = Λ(x)dx. (2.45) σ E D,σ Then we get that the local matrices (A σσ ) σσ E are symmetric and positive, and the identification of the numerical fluxes using (2.37) leads to the expression: F,σ (u) = (u u σ ). (2.46) A σσ σ E The properties provided by this definition (which could not have been obtained using natural expansions of regular functions) are shown in Lemma 2.8. Then Theorem 2.1 shows that these properties are sufficient to provide the convergence of the scheme. Note that the proof of this property holds for general heterogeneous, anisotropic and possibly discontinuous fields Λ, for which the solution u of (2.5) is not in general more regular than u H 1 0 (). The local consistency property provided by definition (2.46) is only detailed in the error estimate theorem 2.2, in the case where Λ and u are regular enough. Remark 2.8 The choice of the coefficient d in (2.41) is not compulsory, and any fixed positive value could be substituted; it is motivated by the fact that it provides a diagonal matrix A in the case of isotropic diffision and of meshes which satisfy n,σ = x σ x d,σ (triangular, rectangular, orthogonal parallelepipedic meshes but unfortunately not general tetrahedric meshes), and yields the usual two point scheme. In this case, the formula (2.38) leads to the discrete gradient which was introduced in [22]. Mixed-type schemes on general non matching grids We again consider that D = (M, E, P) is only a discretization of in the sense of definition 1.2, and we then denote by x σ the center of gravity of σ E. Our purpose is again to define a relation between the numerical fluxes F,σ (u) and the values u, (u σ ) σ E. But the idea is now, defining the space of the numerical fluxes Y D to connect an element F Y D with an element u X D,0 by F, G Y = Y D = {((F,σ ) σ E ) M, F,σ R}. (2.47) M σ E (u u σ )G,σ, G Y D, (2.48) through a bilinar form, Y on Y D. This bilinear form allows to consider the matrix M defined by M σσ = 1,σ, 1,σ Y, and then the bilinear form can be rewritten in the equivalent form F, G Y = M σσ M σ E σ E F,σ G,σ. The scheme is then expressed by the relations (2.8)-(2.9)-(2.10) in addition to (2.48). Hence we must now define a suitable bilinear form on Y D (this problem is explored by the mixed finite element method or the mimetic finite difference method [6]). Instead of defining general orthogonality properties in the space Y D, we here consider an idea similar to (2.40). Let us define, in the spirit of (2.39), m()λ v (F ) = σ E F,σ (x σ x ), (2.49) with Λ = 1 Λ(x)dx. m()

19 18 Note that the definition F, G Y = M m()λ v (F ) v (G) leads to a non-invertible matrix. We then introduce the linear form and H,σ (F ) = F,σ m(σ) Λ v (F ) n,σ, (2.50) v,σ (F ) = v (F ) + dh,σ (F )Λ 1 x σ x. (2.51) d,σ We may then define v D F as the piecewise constant function equal to v,σ (F ) a.e. in the cone D,σ with vertex x and basis σ: v D F (x) = v,σ (F ) for a.e. x D,σ, (2.52) and we set, following the same idea as (2.37) F, G Y = Λ(x)v D F (x) v D G(x)dx. (2.53) Remark 2.9 As we noticed for the gradient scheme, the choice of the coefficient d in (2.51) is not compulsory, and any fixed positive value could be substituted; it is again motivated by the fact that it provides a diagonal matrix M in the case of isotropic diffision and of meshes which satisfy n,σ = x σ x d,σ (triangular, rectangular, orthogonal parallelepipedic meshes but unfortunately not general tetrahedric meshes), and yields the usual two point scheme. In this case, the formula (2.38) leads to the discrete gradient which was introduced in [22]. Remark 2.10 Let us consider the case where Λ is constant in each control volume M. Note that, for a slightly different definition of v D, we may find exactly a mixed finite element method. Indeed, let us define and This definition ensures the property v,σ (F )(x) = v (F ) + H,σ (F )Λ 1 x x, x D,σ, (2.54) d,σ v D F (x) = v,σ (F )(x) for a.e. x D,σ, M, σ E. (2.55) Λ v,σ (F )(x) n,σ = F,σ m(σ), x σ, σ E, M. Therefore, under the condition (2.10), we get that Λ v D F Hdiv() (it is easy to see the continuity of the normal component on the boundary of each cone D,σ ). Defining V D = {Λ v D F, F Y D such that (2.10) holds }, the scheme (2.8)-(2.9)-(2.10), in addition to (2.53) with (2.54) resumes to a classical mixed finite element method, with an explicit basis for the fluxes, different from the Raviart-Thomas one on triangles or rectangles, since it can be written find g V D and u H M () such that g(x) Λ 1 (x)ĝ(x)dx = u(x)divĝ(x)dx, ĝ V D, (2.56) and û(x)divg(x)dx = û(x)f(x)dx, û H M (). (2.57)

20 Numerical results We present some numerical results obtained with various choices of A in the scheme (2.28),(2.12) with the flux (2.46), which we synthetize here for the sake of clarity: Find u X D,A (that is (u ) M, (u σ ) σ Ehyb ), such that: F,σ (u)(v v σ ) = v f(x)dx, for all v X D,A, M σ E M with F,σ (u) = (u σ u ), M, σ E. A σσ σ E (2.58) Order of convergence We consider here the numerical resolution of Equation (2.1) supplemented by a homogeneous Dirichlet boundary condition; the right hand side is chosen so as to obtain an exact solution to the problem, so as to easily compute the error between the exact and approximate solutions. We consider Problem (2.1) with a constant matrix Λ: Λ = ( ), (2.59) and choose f : R and f such that the exact solution to Problem (2.1) is ū : R defined by ū(x, y) = 16x(1 x)y(1 y) for any (x, y). Note that in this case, the composite scheme is in fact the cell centred scheme, there are no edge unknowns. (2.24). Let us first consider conforming meshes, such as the triangular meshes which are depicted on Figure 2.2 and uniform square meshes. Figure 2.2: Regular conforming coarse and fine triangular grids For both A = (hybrid scheme) and A = E int (cell centred scheme), the order of convergence is close to 2 for the unknown u and 1 for its gradient. Of course, the hybrid scheme is almost three times more costly in terms of number of unknowns than the cell centred scheme for a given precision. However, the number of nonzero terms in the matrix is, again for a given precision on the approximate solution, larger for the cell centred scheme than for the hybrid scheme. Hence the number of unknowns is probably not a sufficient criterion for assessing the cost of the scheme. Results were also obtained in the case of uniform square or rectangular meshes. They show a better rate of convergence of the gradient (order 2 in the case of the hybrid scheme and 1.5 in the case of the centred scheme), even though the rate of convergence of the approximate solution remains unchanged and close to 2. We then use a rectangular nonconforming mesh, obtained by cutting the domain in two vertical sides and using a rectangular grid of 3n 2n (resp. 5n 2n) on the first (resp. second side), where n is the number of the mesh, n = 1, Again, the order of convergence which we obtain is 2 for u and around 1.8 for the gradient. We give in Table 2.1 below the errors obtained in the discrete L 2 norm for u and u for a nonconforming mesh and (in terms of number of unknowns) and for the rectangular 4 6 and 4 10 conforming rectangular meshes, for both the hybrid and cell centred schemes. We show on Figure 2.3 the solutions for the corresponding grids (which looks very much the same for both schemes).

21 20 NU NM ɛ(u) ɛ( u) n Hyb Cent Hyb Cent Hyb Cent Hyb Cent C E E E E-02 NC E E E E-02 C E E E E-02 Table 2.1: Error for the non conforming rectangular mesh, hybrid scheme (Hyb) and centred (Cent) schemes. For both schemes: NU is the number of unknowns in the resulting linear system, NM the number of non zero terms in the matrix, ɛ(u) the discrete L 2 norm of the error on the solution and ɛ( u) the discrete L 2 norm of the error on the gradient. C1 and C2 are the two conforming meshes represented on the left and the right in Figure 2.3, and NC the non conforming one represented in the middle. Figure 2.3: The approximate solution for conforming and nonconforming meshes. Left: conforming 8 6 mesh, center: non conforming 4 6, 4 10 mesh, right: conforming Further detailed results on several problems and conforming, non conforming and distorted meshes may be found in [23]. The case of a highly heterogeneous tilted barrier We now turn to a heterogeneous case. The domain =]0, 1[ ]0, 1[ is composed of 3 subdomains, which are depicted in Figure 2.4: 1 = {(x, y) ; ϕ 1 (x, y) < 0}, with ϕ 1 (x, y) = y δ(x.5).475, 2 = {(x, y) ; ϕ 1 (x, y) > 0, ϕ 2 (x, y) < 0}, with ϕ 2 (x, y) = ϕ 1 (x, y) 0.05, 3 = {(x, y) ; ϕ 2 (x, y) > 0}, and δ = 0.2 is the slope of the drain (see Figure 2.4). Dirichlet boundary conditions are imposed by setting the boundary values to those of the analytical solution given by u(x, y) = ϕ 1 (x, y) on 1 3 and u(x, y) = ϕ 1 (x, y)/10 2 on 2. The permeability tensor Λ is heterogeneous and isotropic, given by Λ(x) = λ(x)id, with λ(x) = 1 for a.e. x 1 3 and λ(x) = 10 2 for a.e. x 2. Note that the isolines of the exact solution are parallel to the boundaries of the subdomain, and that the tangential component of the gradient is 0. We use the meshes depicted in figure 2.4. Mesh 3 (containing control volumes) is obtained from Mesh 1 by the addition of two layers of very thin control volumes around each of the two lines of discontinuity of Λ: because of the very low thickness of these layers, equal to 1/10000, the picture representing Mesh 3 is not different from that of Mesh 1. We get the following results for the approximations of the four fluxes at the boundary. nb. unknowns matrix size x = 0 x = 1 y = 0 y = 1 analytical mesh centred mesh mesh composite mesh mesh hybrid mesh mesh

22 Figure 2.4: Domain and meshes used for the tilted barrier test: mesh 1 (10 21 center), mesh 2 ( right) Note that the values of the numerical solution given by the hybrid and composite schemes are equal to those of the analytical one (this holds under the only condition that the interfaces located on the lines ϕ i (x, y) = 0, i = 1, 2 are not included in A, and that, for all σ A, all M with β σ 0 are included in the same subdomain i ). Note that Mesh 3, which leads to acceptable results for the computation of the fluxes, is not well designed for such a coupled problem, because of too small measures of control volumes. Hence the composite method on Mesh 1 appears to be the most suitable method for this problem. Tilted barrier The domain =]0, 1[ ]0, 1[ is composed of 3 subdomains: 1 = {(x, y) ; ϕ 1 (x, y) < 0}, with ϕ 1 (x, y) = y δ(x.5).475, 2 = {(x, y) ; ϕ 1 (x, y) > 0, ϕ 2 (x, y) < 0}, and ϕ 2 (x, y) = ϕ 1 (x, y) 0.05, 3 = {(x, y) ; ϕ 2 (x, y) > 0}, and δ = 0.2 is the slope of the drain. We impose, at the boundary of the domain, the values of the analytical solution given by u(x, y) = ϕ 1 (x, y) on 1, u(x, y) = ϕ 1 (x, y)/10 2 on 2, u(x, y) = ϕ 2 (x, y) 0.05/10 2 on 2. The permeability tensor Λ is heterogeneous and isotropic, given by Λ(x) = λ(x)id, with λ(x) = 1 for a.e. x 1, λ(x) = 10 2 for a.e. x 2 and λ(x) = 1 for a.e. 3. Mesh 3 (containing control volumes) is obtained from Mesh 1 by the addition of two layers of very thin control volumes around each of the two lines of discontinuity of Λ: because of the very low thickness of these layers, equal to 1/10000, the picture representing Mesh 3 is not different from that of Mesh 1. We get the following results for the approximations of the four fluxes at the boundary. nb. unknowns matrix size x = 0 x = 1 y = 0 y = 1 analytical E bar = E int (mesh 1) E bar = E int (mesh 2) E bar = E int (mesh 3) E bar E int (mesh 1) E bar = (mesh 1) Note that in the cases E bar E int and E bar =, the numerical solution is equal to the analytical one (this holds under the only conditions that the interfaces located on the lines ϕ i (x, y) = 0, i = 1, 2 are not included in E bar, and that, for all σ E bar, all N σ are included in the same subdomain i ). Note that Mesh 3, which leads to acceptable results for the computation of the fluxes, is not well designed for such a coupled problem, because of too small measures of control volumes. Hence the partly hybrid method appears to be the most suitable method for this problem and this mesh. 2.3 Further analysis We now give the abstract properties of the discrete fluxes which are sufficient to prove the convergence of the schemes provided in this chapter. Let us first introduce some notations related to the mesh. Let D = (M, E, P) be

23 22 a discretization of in the sense of definition 1.2. The size of the discretization D is defined by: h D = sup{diam(), M}, and the regularity of the mesh by: ( d,σ θ D = max max, σ E int,,l M σ d L,σ max h M,σ E d,σ ). (2.60) For a given set A E int and for a given family (βσ ) M satisfying property (2.25), we introduce some measure σ E int of the resulting regularity with ( ) L M θ D,A = max θ D, max βl σ x L x σ 2. (2.61) M,σ E A Definition 2.2 (Continuous, coercive, consistent and symmetric families of fluxes) Let F be a family of discretizations in the sense of definition 1.2. For D = (M, E, P) F, M and σ E, we denote by F,σ D a linear mapping from X D to R, and we denote by Φ = ((F,σ D ) M ) D F. We consider the bilinear form defined by u, v F = M σ E h 2 σ E F D,σ(u)(v v σ ), (u, v) X 2 D. (2.62) The family of numerical fluxes Φ is said to be continuous if there exists M > 0 such that u, v F M u X v X, (u, v) X 2 D, D = (M, E, P) F. (2.63) The family of numerical fluxes Φ is said to be coercive if there exists α > 0 such that α u 2 X u, u F, u X D, D = (M, E, P) F. (2.64) The family of numerical fluxes Φ is said to be consistent (with Problem (2.1)) if for any family (u D ) D F satisfying: u D X D,0 for all D F, there exists C > 0 with u D X C for all D F, there exists u L 2 () with lim h D 0 Π Mu D u L 2 () = 0 (we get from Lemma 5.7 that u H 1 0 ()), then the following holds lim u D, P D ϕ F = h D 0 Finally the family of numerical fluxes Φ is said to be symmetric if We may now state the general convergence theorem. Λ(x) ϕ(x) u(x)dx, ϕ C c (). (2.65) u, v F = v, u F, (u, v) X 2 D, D = (M, E, P) F. Theorem 2.1 Let F be a family of discretizations in the sense of definition 1.2; for any D F, let A E int and (βσ ) M satisfying property (2.25). We assume that there exists θ > 0 such that θ D,A θ, for all D F, where σ E int θ D,A is defined by (2.61). Let Φ = ((F,σ D ) M ) D F be a continuous, coercive and symmetric and consistent σ E family of numerical fluxes in the sense of definition 2.2. Let (u D ) D F be the family of functions solution to (2.28) for all D F. Then Π M u D converges in L 2 () to the unique solution u of (2.5) as h D 0. Moreover D u D converges to u in L 2 () d as h D 0.

24 23 Proof. We let v = u D in (2.28), we apply the Cauchy-Schwarz inequality to the right hand side. We get u D, u D D = f(x)π M u D (x)dx f L 2 () Π M u D L 2 (). We apply the Sobolev inequality (5.13) with p = 2, which gives in this case Π M u D L 2 () C 1 Π M u 1,2,M. Using (2.84) and the consistency of the family Φ of fluxes, we then have This leads to the inequality α Π M u D 2 X C 1 f L 2 () u D X. u 1,2,M u D X C 1 α f L 2 (). (2.66) Thanks to lemma 5.7, we get the existence of u H0 1 (), and of a subfamily extracted from F, such that Π M u D u L 2 () tends to 0 as h D 0. For a given ϕ Cc (), let us take v = P D,A ϕ in (2.28) (recall that P D,A ϕ X D,A ). We get u D, P D,A ϕ F = f(x)p M ϕ(x)dx. Let us remark that, thanks to the continuity of the family Φ of fluxes, we have u D, P D,A ϕ P D ϕ F M C 1 α f L 2 () P D,A ϕ P D ϕ X. Thanks to (2.25) and (2.61), we get the existence of C ϕ, only depending on ϕ such that, for all M and all σ A E, βσ L ϕ(x L ) ϕ(x σ ) βσ L x L x σ 2 C ϕ θ D,A C ϕ h 2. (2.67) We can then deduce L M L M lim P D,Aϕ P D ϕ X = 0. (2.68) h D 0 Thanks to the properties of subfamily extracted from F, we can apply the consistency hypothesis on the family Φ of fluxes, which gives lim u D, P D ϕ F = Λ(x) ϕ(x) u(x)dx. h D 0 Gathering the two above results leads to lim u D, P D,A ϕ F = Λ(x) ϕ(x) u(x)dx, h D 0 which concludes the proof that Λ(x) ϕ(x) u(x)dx = f(x)ϕ(x)dx. Therefore, u is the unique solution of (2.5), and we get that the whole family (u D ) D F converges to u as h D 0. Let us now prove the second part of the theorem. Let ϕ C c () be given (this function is devoted to approximate u in H0 1 ()). Thanks to the Cauchy-Schwarz inequality, we have D u D (x) u(x) 2 dx 3 (T1 D + T2 D + T 3 ) where T1 D = D u D (x) D P D ϕ(x) 2 dx,

25 and We have, thanks to Lemma 2.7, T2 D = T 3 = D P D ϕ(x) ϕ(x) 2 dx, ϕ(x) u(x) 2 dx. 24 lim T 2 D = 0. (2.69) h D 0 We have, thanks to Lemma 2.5 and to the coercivity of the family of fluxes, that there exists C 2 such that with C 2 = C2 12 α. Taking v = u D P D ϕ, we have D v 2 L 2 () d C2 12 v 2 X C 2 v, v F, v X D, T D 1 C 2 ( u D, u D F 2 u D, P D ϕ F + P D ϕ, P D ϕ F ). Using the result of convergence proved for u D and the consistency of the family of fluxes, we get lim u D, P D ϕ F = u(x) Λ(x) ϕ(x)dx. (2.70) h D 0 The fact that P D ϕ X remains bounded, results from the regularity of ϕ and the regularity hypotheses of the family of discretizations. Hence we can use the consistency of the family of fluxes, which writes in this case lim P Dϕ, P D ϕ F = ϕ(x) Λ(x) ϕ(x)dx. (2.71) h D 0 Remarking that passing to the limit h D 0 in (2.28) with v = u D provides that u D, u D F converges to u Λ udx, we get that lim u D P D ϕ, u D P D ϕ F = (u ϕ) Λ (u ϕ)dx λ u ϕ 2 dx, h D 0 which yields lim sup T1 D h D 0 C 2 λ u ϕ 2 dx. From the above results, we obtain that there exists C 3, independent of D, such that D u D (x) u(x) 2 dx C 3 ϕ(x) u(x) 2 dx + T4 D, with (noting that ϕ is fixed) lim T 4 D = 0. (2.72) h D 0 Let ε > 0. We can choose ϕ such that ϕ(x) u(x) 2 dx ε, and we can then choose h D small enough such that T4 D ε. This completes the proof that lim D u D (x) u(x) 2 dx = 0 (2.73) h D 0 in the case of a general continuous, coercive, consistent and symmetric family of fluxes. Let us write an error estimate, in the particular case that Λ = Id and that the solution of (2.5) is regular enough.

26 25 Theorem 2.2 (Error estimate, isotropic case) We consider the particular case Λ = Id, and we assume that the solution u H 1 0 () of (2.5) is in C 2 (). Let D = (M, E, P) be a discretization in the sense of definition 1.2, let A E int be given, let A = (β σ ) σ A, M be a family of real numbers such that (2.25) holds, and let θ θ D,A be given (see (2.61)). Let (F,σ ) M,σ E be a family of linear mappings from X D to R, such that there exists α > 0 with α u 2 X u, u F, u X D, (2.74) defining u, v F by (2.62). We denote by Ru = ( M σ E ( ) 2 1/2 d,σ F,σ (P D,A u) + u(x) n,σ dγ(x)). (2.75) m(σ) σ Then the solution u D of (2.28) verifies that there exists C 4, only depends on α and on θ, such that and verifies that there exists C 5, only depending on α, θ and u such that Π M u D P M u L 2 () C 4 Ru, (2.76) D u D u L 2 () d C 5 (Ru + h D ). (2.77) Remark 2.11 The extension of Theorem 2.2 to the case u H 2 () could be studied in the case d = 2 or d = 3. Such a study, which demands a rather longer and more technical proof, is not expected to provide more information on the link between accuracy and the regularity of the mesh than the result presented here. Proof. Let v X D, since u = f, we get: v u(x)dx = M f(x)π M v(x)dx. (2.78) Thanks to the following equality (recall that u C 2 () and therefore u n,σ is defined on each edge σ) v u(x)dx = (v v σ ) u(x) n,σ dγ(x), M M σ E σ we get that P D,A u, v F = f(x)π M v(x)dx + M σ E ( ) F,σ(P D D,A u) + u(x) n,σ dγ(x) (v v σ ). σ Taking v = P D,A u u D X D,A in this latter equality and using (2.78) we get v, v F = ( ) F,σ(P D D,A u) + u(x) n,σ dγ(x) (v v σ ), M σ E which leads, using (2.74) and the Cauchy-Schwarz inequality, to σ α v X Ru. (2.79) Using (2.84) and the Sobolev inequality (5.13) with p = 2 provides the conclusion of (2.76). Let us now prove (2.77). We have D u D u L 2 () d Du D D P D,A u L 2 () d + DP D,A u u L 2 () d. The bound of the first term in the above right hand side is bounded thanks to Lemma 2.5 and (2.79). The inequality D P D,A u u L 2 () d C 6h D is obtained thanks to Lemma 2.7 and using a similar inequality to (2.67), replacing ϕ by u.

27 The convergence proof in the (harmonic averaging) isotropic case Lemma 2.1 [Properties of the numerical fluxes in the harmonic averaging scheme for isotropic diffusion, adapted mesh] Under hypothesis (1.5) and (2.19), let F be a family of pointed adapted polygonal finite volume space discretization of in the sense of Definition 1.4. Let Φ = ((F,σ D ) M ) D F be defined by (2.20). Then Φ is a continuous, coercive and symmetric and consistent family of numerical fluxes in the sense of definition 2.2. Proof. The only nonobvious property is the consistency of the family of numerical fluxes. Let (u D ) D F be a family satisfying: u D X D,0 for all D F, there exists C > 0 with u D X C for all D F, there exists u L 2 () with lim h D 0 Π Mu D u L 2 () = 0 (we get from Lemma 5.7 that u H 1 0 ()), For any D F, we define the function G D L D () d (see Definition 1.3), by σ E G D,σ = d u σ u d,σ n,σ, M, σ E. (2.80) Let us show that G D weakly converges to u in L 2 () d as h(m) 0. We first prolong u and G D by 0 outside. Let us first remark that, thanks to the Cauchy-Schwarz inequality and using m(d,σ ) = m(σ)d,σ /d, G D 2 L 2 (R d ) d = d M σ E m(σ)d,σ ( uσ u d,σ ) 2 = d u D 2 X. Since G D L2 (R d ) d remains bounded, we can extract from F a subfamily, again denoted F, such that there exists G L 2 (R d ) d and G D weakly converges in L 2 (R d ) d to G as h(m) 0. Let us prove that G = u, which implies, by uniqueness of the limit, that the whole family converges to u as h(m) 0. Let ϕ Cc 1 () d be given. Let us denote by ϕ D L D () d 1 the element defined by the value m(σ) σ ϕ(x)dγ(x), in D,σ for all M σ, again prolonged by 0 outside. We then have that ϕ D converges to ϕ in L 2 (R d ) d. We 1 get, noticing that the terms u σ vanish for the interior edges since m(σ) σ ϕ(x)dγ(x) n,σ + 1 m(σ) σ ϕ(x)dγ(x) n L,σ = 0, that R d G D (x) ϕ(x)dx = M u σ E σ ϕ(x)dγ(x)n,σ = u D (x)div(ϕ)(x)dx, R d which implies, by passing to the limit h(m) 0, G(x) ϕ(x)dx = u(x)div(ϕ)(x)dx. R d R d This proves that u H 1 (R d ). Since u = 0 outside, we get that u H0 1 (R d ), and that G(x) ϕ(x)dx = u(x) ϕ(x)dx, R d R d which proves that G = u. For any D F, we now define the function H D L D () d (see Definition 1.3), by H D,σ = ϕ(x σ) ϕ(x ) d σ n,σ + ϕ(x ) ( ϕ(x ) n,σ )n,σ, M, σ E. (2.81)

28 27 Then H D converges to ϕ in L () d as h(m) 0. Indeed, we have, for M and σ E, using the choice of x σ for adapted pointed finite volume discretizations, ϕ(x σ ) ϕ(x ) = d,σ ϕ(x ) n,σ + d2,σ 2 R,σ, with R,σ is bounded by some L norm of the second order partial derivatives of ϕ, denoted by C 7. We consider h(m) small enough, such that ϕ vanishes on all M such that E E ext, hence for σ E E ext, We then get Using that H D,σ = 0 = ϕ(x ). H D,σ ϕ(x ) d σ 2 C 7 h(m)c 7, M, σ E. u D, P D ϕ X = λ D (x)g D (x) H D (x)dx, in which λ D (x) denotes the piecewise function equal to λ in each M, we get, by weak-strong convergence, lim u D, P D ϕ X = λ(x) u(x) ϕ(x)dx. h(m) 0 Hence the conclusion of the consistency of the family of numerical fluxes. We then get the convergence of the scheme and an error estimate, using Theorem 2.1 with A = The convergence proof in the case of the gradient schemes Lemma 2.2 [Properties of the numerical fluxes in the gradient scheme on adapted meshes for anisotropic diffusion] Under hypothesis (1.5) and (2.3), let F be a family of pointed adapted polygonal finite volume space discretization of in the sense of Definition 1.4. Let Φ = ((F,σ D ) M ) D F be defined by (2.33). Then Φ is a continuous, coercive and symmetric and consistent family of numerical fluxes in the sense of definition 2.2. Proof. get Hence Let us first show that this family is coercive. We apply (2.88) to (R,σ u) 2, defining β > 0 below. We Choosing β = (R,σ u) 2 α d β 1 + β ( Λ(x) D u(x) D u(x) + R D u(x) 2) dx M λ α d ( σ E ( ) 2 uσ u α dβ u 2. (2.82) d,σ (λ α dβ )m() u 2 + α β 1 + β 2ab and using a+b min(a, b) for a, b > 0, we get σ E m(σ)d,σ ( ) ) 2 uσ u d,σ ( Λ(x) D u(x) D u(x) + R D u(x) 2) dx min( 1 ( ) 2 uσ u λ, α) m(σ)d,σ = min( 1 d d,σ d λ, α) u D 2 X. M σ E Hence we get the coerciveness property. The consistency property is resulting from the weak and strong convergence properties (Lemmas 2.3 and 2.4) of D u defined by (2.30). Let us state and prove the needed results on the discrete gradient defined by (2.30).

29 28 Lemma 2.3 Let F be a family of discretizations in the sense of definition 1.4 such that there exists θ > 0 with θ θ D for all D F. Let (u D ) D F be a family of functions, such that: u D X D,Eint for all D F, there exists C > 0 with u D X C for all D F, there exists u L 2 () with lim h D 0 Π Mu D u L 2 () = 0, Then u H 1 0 () and D u D weakly converge in L 2 () d to u as h D 0, where the operator D is defined by (2.30). Proof. Let us prolong Π M u D and D u D by 0 outside of. Thanks to the Cauchy-Schwarz inequality, we easily get that D u D 2 L 2 () d d u D 2 X. Hence, up to a subsequence, there exists some function G L 2 (R d ) d such that D u D weakly converges in L 2 (R d ) d to G as h D 0. Let us show that G = u. Let ψ Cc (R d ) d be given. We assume that h(m) is small enough for ensuring that the support of ψ does not intersect any boundary control volume. Let us consider the term T5 D defined by T5 D = D u D (x) ψ(x)dx. R d Using x L x = d σ n,σ, we get that T D 5 = T D 6 + T D 7, with and T D 6 = 1 2 σ E int M σ ={,L} T D 7 m(σ)(u L u )n,σ (ψ + ψ L ), with ψ = 1 m() = We compare T D 6 with T D 8 defined by with We get that (T D 6 T D 8 ) 2 M σ E E int M σ={,l} T D 8 = σ E int M σ ={,L} σ E int M σ ={,L} m(σ) u σ u d,σ (x σ x ) (ψ ψ L ). m(σ)(u L u )n,σ ψ σ, ψ σ = 1 ψ(x)dγ(x). m(σ) σ m(σ) d σ (u L u ) 2 M m(σ)d,σ ψ + ψ L 2 σ E which leads to lim (T 6 D T8 D ) = 0. h D 0 Since T8 D = m(σ)u n,σ ψ σ = Π M u D (x)divψ(x)dx, M σ E R d ψ(x)dx, ψ σ 2, we get that lim T 8 D = u(x)divψ(x)dx. Let us now turn to the study of T D h D 0 R d 7. We have, using the Cauchy- Schwarz inequality and the regularity of the mesh (T D 7 ) 2 C 8 u 2 Xh(M) 2,

30 29 where C 8 only depends on ϕ and θ. Hence lim T 7 D = 0. h D 0 This proves that the function G L 2 (R d ) d is a.e. equal to u in R d. Since u = 0 outside of, we get that u H 1 0 (), and the uniqueness of the limit implies that the whole family D u D weakly converges in L 2 (R d ) d to u as h D 0. Let us now state some strong consistency property of the discrete gradient applied to the interpolation of a regular function. Lemma 2.4 Let D be a discretization of in the sense of Definition 1.4, and let θ θ D be given. Then, for any function ϕ C 2 (), there exists C 9 only depending on d, θ and ϕ such that: where D is defined by (2.30). D P D ϕ ϕ (L ()) d C 9h D, (2.83) Proof. Taking into account definition (2.30) and (2.39), we write, for any M, P D ϕ ϕ(x ) = 1 ( ) ϕ(xσ ) ϕ(x ) m(σ) ϕ(x ) n,σ (x σ x ). m() d,σ σ E Thanks to ϕ(x σ ) ϕ(x ) ϕ(x ) n,σ C 10h(M), d,σ where C 10 only depends on ϕ, we conclude the proof. We then get the convergence of the scheme and an error estimate, using Theorem 2.1 with A = E int. Convergence study in the case of general meshes In this section, we denote by x σ the center of gravity of σ E. For all v X D, we denote by Π M v H M () the piecewise function from to R defined by Π M v(x) = v for a.e. x, for all M. Using the Cauchy-Schwarz inequality, we have for all σ E int with M σ = {, L}, which leads to the relation (v v L ) 2 d σ (v v σ ) 2 d,σ + (v σ v L ) 2 d L,σ, v X D, Π M v 2 1,2,M v 2 X, v X D,0. (2.84) The following lemma provides an equivalence property between the L 2 -norm of the discrete gradient, defined by (2.38)-(2.42) and the norm X. Lemma 2.5 Let D be a discretization of in the sense of Definition 1.3, and let θ θ D be given (where θ D is defined by (2.60)). Then there exists C 11 > 0 and C 12 > 0 only depending on θ and d such that: where D is defined by (2.38)-(2.42). C 11 u X D u L 2 () C 12 u X, u X D, (2.85) Proof. By definition, D u 2 L 2 () = d M m(σ)d,σ,σ u 2. d σ E

31 30 From the definition (2.41), thanks to (2.39) and to the definition (2.38), we get that Therefore, m(σ)d,σ R,σ u n,σ = 0, M. (2.86) d σ E D u 2 L 2 () = ( m() d u 2 + ) m(σ)d,σ (R,σ u) 2. (2.87) d M σ E Let us now notice that the following inequality holds: (a b) 2 λ 1 + λ a2 λb 2, a, b R, λ > 1. (2.88) We apply this inequality to (R,σ u) 2 for some λ > 0 and obtain: This leads to Choosing λ as we get that m(σ)d,σ d σ E (R,σ u) 2 λd 1 + λ (R,σ u) 2 λ 1 + λ ( ) 2 ( ) 2 uσ u λd u 2 xσ x. (2.89) d,σ σ E m(σ)d,σ d,σ ( ) 2 uσ u λm()d u 2 θ 2. d,σ λ = 1 dθ 2, (2.90) D u 2 (L 2 ()) d λ 1 + λ u 2 X, which shows the left inequality of (2.85). Let us now prove the right inequality. Remark that, thanks to the assumption that is x -star-shaped, the property σ E m(σ)d,σ = d m(), M (2.91) holds. On one hand, using the definition (2.38) of u and (2.91), the Cauchy Schwarz inequality leads to: u 2 1 ( m(σ) m() )2 (u σ u ) 2 m(σ)d,σ = d,σ σ E σ E d m() m(σ) (u σ u ) 2. (2.92) d,σ σ E On the other hand, by definition (2.41), and thanks to the regularity of the mesh (2.95), we have: ( (R,σ u) 2 2d ( u σ u ) 2 + u 2 x ) ( σ x 2 2d ( u ) σ u ) 2 + θ 2 u 2. (2.93) d,σ d,σ d,σ From (2.87), (2.92) et (2.93), we conclude that the right inequality of (2.85) holds. We can now state a result of weak convergence for the discrete gradient of a sequence of bounded discrete functions. Lemma 2.6 Let F be a family of discretizations in the sense of definition 1.3 such that there exists θ > 0 with θ θ D for all D F. Let (u D ) D F be a family of functions, such that: u D X D,0 for all D F,

32 31 there exists C > 0 with u D X C for all D F, there exists u L 2 () with lim h D 0 Π Mu D u L 2 () = 0, Then u H 1 0 () and D u D weakly converge in L 2 () d to u as h D 0, where the operator D is defined by (2.38)-(2.42). Remark 2.12 Note that the proof that u H 1 0 () also results from (2.84), which allows to apply Lemma 5.7 of the appendix in the particular case p = 2. Proof. Let us prolong Π M u D and D u D by 0 outside of. Thanks to Lemma 2.5, up to a subsequence, there exists some function G L 2 (R d ) d such that D u D weakly converges in L 2 (R d ) d to G as h D 0. Let us show that G = u. Let ψ Cc (R d ) d be given. Let us consider the term T9 D defined by T9 D = D u D (x) ψ(x)dx. R d We get that T D 9 = T D 10 + T D 11, with T D 10 = M σ E m(σ)(u σ u )n,σ ψ, with ψ = 1 m() ψ(x)dx, and T11 D = R,σ u n,σ ψ(x)dx. M σ E D,σ We compare T10 D with T12 D defined by T12 D = m(σ)(u σ u )n,σ ψ σ, M σ E with We get that (T D 10 T D 12) 2 M ψ σ = 1 ψ(x)dγ(x). m(σ) σ m(σ) (u σ u ) 2 d,σ σ E M σ E m(σ)d,σ ψ ψ σ 2, which leads to lim (T 10 D T12) D = 0. h D 0 Since T12 D = m(σ)u n,σ ψ σ = Π M u D (x)divψ(x)dx, M σ E R d we get that lim T 12 D = u(x)divψ(x)dx. Let us now turn to the study of T h D 0 R 11. D Noting again that (2.86) d holds, we have: T11 D = R,σ u n,σ (ψ(x) ψ )dx. M σ E D,σ Since ψ is a regular function, there exists C ψ only depending on ψ such that D,σ (ψ(x) ψ )dx m(σ)d,σ C ψ h D. From (2.93) and the Cauchy-Schwarz inequality, we thus get: d lim T 11 D = 0. h D 0

33 32 This proves that the function G L 2 (R d ) d is a.e. equal to u in R d. Since u = 0 outside of, we get that u H 1 0 (), and the uniqueness of the limit implies that the whole family D u D weakly converges in L 2 (R d ) d to u as h D 0. Let us now state some strong consistency property of the discrete gradient applied to the interpolation of a regular function. Lemma 2.7 Let D be a discretization of in the sense of Definition 1.3, and let θ θ D be given. Then, for any function ϕ C 2 (), there exists C 13 only depending on d, θ and ϕ such that: where D is defined by (2.38)-(2.42). D P D ϕ ϕ (L ()) d C 13h D, (2.94) Proof. Taking into account definition (2.42), and using definition (2.40), we write: From (2.38), we have, for any M,,σ P D ϕ ϕ(x ) P D ϕ ϕ(x ) + R,σ P D ϕ P D ϕ = = 1 m() 1 m() m(σ)(ϕ(x σ ) ϕ(x )) σ E ( m(σ) ϕ(x ) (x σ x ) + h 2 ) ρ,σ n,σ, σ E where ρ,σ C ϕ with C ϕ only depending on ϕ. Thanks to (2.39) and to the regularity of the mesh, we get: P D ϕ ϕ(x ) 1 m(σ)h 2 m() ρ,σ h d C ϕ θ. σ E From this last inequality, using Definition 2.41, we get: d R,σ P D ϕ = ϕ(x σ ) ϕ(x ) P D ϕ (x σ x ) d,σ d ( h 2 d ρ,σ + h 2 d C ϕ θ ),σ dθ(h C ϕ + h dc ϕ θ), which concludes the proof. Lemma 2.8 Let F be a family of discretizations in the sense of definition 1.3. We assume that there exists θ > 0 with θ D θ, D F, (2.95) where θ D is defined by (2.60). Let Φ = ((F D,σ ) M D σ E ) D F be the family of fluxes defined by (2.43)-(2.46). Then the family Φ is a continuous, coercive, consistent and symmetric family of numerical fluxes in the sense of definition 2.2. Proof. Since the family of fluxes is defined by (2.43)-(2.46), it satisfies (2.37), and therefore we have: u, v F = D u(x) Λ(x) D v(x)dx, u, v X D. Hence the property u, v F = v, u F holds. The continuity and coercivity of the family Φ result from Lemma 2.5 and the properties of Λ, which give: u, v F λ D u L 2 () D v L 2 () and u, v F λ D u 2 L 2 () for any u, v X D. The consistency results from the weak and strong convergence properties of lemmas 2.6 and 2.7, which give D u D u weakly in L 2 () and D P D ϕ ϕ in L 2 () as the mesh size tends to 0. Let us apply the error estimate, in the particular case that Λ = Id and that the solution of (2.5) is regular enough.

34 33 Theorem 2.3 (Error estimate, isotropic case) Under the hypotheses and notations of Theorem 2.2 and in the particular case where (F,σ ) M,σ E is defined by (2.43)-(2.46), there exists C 14, only depending on α, θ and u, such that Ru C 14 h D. (2.96) Proof. Let us assume that the family of fluxes is defined by (2.43)-(2.46). Indeed, we get in this case that, for all v X D, F,σ (v) = ( v + R,σ v n,σ ) m(σ )d,σ y σ σ, d σ E with y σ σ = Using (2.39), we get that ( m(σ) d m() n,σ + 1 m(σ) d,σ m(σ) m() n,σ ) m() n,σ (x σ x ) n,σ if σ = σ d d,σ m() m(σ)n,σ (x σ x )n,σ otherwise. m(σ )d,σ y σ σ = m(σ)n,σ. d σ E Since it is easy to see that there exists C 15 R + such that R,σ P D,A u C 15 h, we then obtain that there exists some C 16 R + with F,σ(P D,A u) + u(x) n,σ dγ(x) C 16m(σ)h. This easily leads to the conclusion of (2.96) The convergence proof in the case of the mixed-type schemes σ Lemma 2.9 Let F be a family of discretizations in the sense of definition 1.3. We assume that there exists θ > 0 such that (2.95) holds, where θ D is again defined by (2.60). Let Φ = ((F,σ D ) M ) D D F be the family of fluxes σ E defined by (2.48)-(2.49)-(2.53). Then the family Φ is a continuous, coercive, consistent and symmetric family of numerical fluxes in the sense of definition 2.2. Proof. Let us remark that we easily get m() v (F ) 2 C 17 m(σ)d,σ σ E Using the Cauchy-Schwarz inequality, we then obtain m(σ)d,σ v,σ (F ) 2 C 18 m(σ)d,σ σ E σ E Hence, letting G,σ = m(σ) d,σ (u u σ ) in (2.48), we get ( ) 2 F,σ. m(σ) ( ) 2 F,σ. m(σ) u 2 X ( F, F Y ) 1/2 ( G, G Y ) 1/2 ( F, F Y C 18 λ) 1/2 u X, which is sufficient to prove the coercivity of the family of fluxes. We now remark that, thanks to (2.39), the property m(σ)d,σ H,σ (F )Λ 1 x σ x = 0 d d,σ σ E

35 34 holds, which allows to write v D F (x) 2 dx = M ( m() v (F ) 2 + m(σ)d,σ d σ E H,σ (F ) 2 Λ 1 x σ x d,σ 2). We then get, using again(2.88) in order to get a lower bound of H,σ (F ) 2, the existence of C 19 such that m() v (F ) 2 C 19 m(σ)d,σ σ E ( ) 2 F,σ, m(σ) which yields the continuity of the family of fluxes (applying the Cauchy-Schwarz inequality to the right-hand side of (2.48)). Note that we shall again get a strong convergence property for v D (F ) do u.

36 Chapter 3 Nonlinear convection - degenerate diffusion problems 3.1 Actual problems and schemes Let be a bounded open subset of R d, (d N ) with boundary and let T R +. One considers the following problem. ( ) u t (x, t) + div q f(u) (x, t) ϕ(u)(x, t) = 0, for (x, t) (0, T ). (3.1) The initial condition is formulated as follows: u(x, 0) = u 0 (x) for x. (3.2) The boundary condition is the following nonhomogeneous Dirichlet condition: u(x, t) = u(x, t), for (x, t) (0, T ). (3.3) This problem arises in different physical contexts. One of them is the problem of two phase flows in a porous medium, such as the air-water flow of hydrological aquifers. In this case, (3.1)-(3.3) represents the conservation of the incompressible water phase, described by the water saturation u, submitted to convective flows (first order space terms q(x, t) f(u)) and capillary effects ( ϕ(u)). The expression q(x, t) f(u) for the convective term in (3.1) appears to be a particular case of the more general expression F (u, x, t), but since it involves the same tools as the general framework, the results of this chapter could be extended to some other problems. One supposes that the following hypotheses, globally referred to in the following as Assumption 3.1, are fulfilled. Assumption 3.1 (H1) We assume hypothesis (1.5) on, (H2) u 0 L () and u L ( (0, T )), u being the trace of a function of H 1 ( (0, T )) L ( (0, T )) (also denoted u); one sets u I = min(infess u 0, infess ū) and u S = max(supess u 0, supess ū), (H3) ϕ is a nondecreasing Lipschitz-continuous function, with Lipschitz constant Φ, and one defines a function ζ such that ζ = ϕ, (H4) f C 1 (R, R), f 0; one sets F = max s [ui,u S ] f (s), (H5) q is the restriction to (0, T ) of a function of C 1 (R d R, R d ), 35

37 36 (H6) div(q(x, t)) = 0 for all (x, t) R d (0, T ), where div(q(x, t)) = of q) and d i=1 q i x i (x, t), (q i is the i-eth component q(x, t).n(x) = 0, for a.e. (x, t) (0, T ), (3.4) (for x, n(x) denotes the outward unit normal to at point x). Remark 3.1 The function f is assumed to be non decreasing in Hypothesis (H3) of Assumption 3.1 for the sake of simplicity. In fact, the convergence analysis which we present here would also hold without this monotonicity assumption using for instance a flux splitting scheme for the treatment of the convective term qf(u). Under Assumption 3.1, (3.1)-(3.3) does not have, in the general case, strong regular solutions. Because of the presence of a non-linear convection term, the expected solution is an entropy weak solution in the sense of Definition 3.2 given below Finite volume approximation We may now define the finite volume discretization of (3.1)-(3.3). Let D be a adapted pointed polygonal finite volume space-time discretization of (0, T ) in the sense of Definitions 1.4 and 1.7. The initial condition is discretized by: In order to introduce the finite volume scheme, we need to define: u 0 = 1 u 0 (x)dx, M. (3.5) m() ū n+1 1 t n+1 σ = δt n ū(x, t)dγ(x)dt, m(σ) t n σ σ E ext, n [[0, N]], (3.6) q n+1,σ = 1 t n+1 δt n q(x, t) n,σ dγ(x)dt, t n σ M, σ E, n [[0, N ]. (3.7) For all n N, u n+1 H M () and g C 0 (R), we introduce the notations u n+1,σ = un+1 L, M, σ E E int, M σ = {, L} u n+1, M, σ E E ext, M σ = {},,σ = ūn+1 σ (3.8) and δ n+1,σ g(u) = g(un+1,σ ) g(un+1 ), M, σ E. (3.9) An implicit finite volume scheme for the discretization of (3.1)-(3.3) is given by the following set of nonlinear equations, the discrete unknowns of which are u = (u n+1 ) M,n [0,N ]: u n+1 δt n un m() + σ E [ (q n+1,σ )+ f(u n+1 ) (qn+1,σ ) f(u n+1,σ ) m(σ) ] δ n+1,σ d ϕ(u) = 0, σ M, n [0, N]], (3.10) where (q n+1,σ )+ and (q n+1,σ ) denote the positive and negative parts of q n+1,σ (i.e. (qn+1,σ )+ = max(q n+1,σ, 0) and (q n+1,σ ) = min(q n+1,σ, 0)).

38 37 Remark 3.2 The upwind discretization of the flux qf(u) in (3.10) uses the monotonicity of f and should be replaced in the general case by, for instance, a flux splitting scheme. Remark 3.3 Thanks to Hypothesis (H6) of Assumption 3.1, one gets for all M and n [[0, N]],,σ =,σ )+ (q n+1,σ ) ] = 0. This leads to σ E q n+1 σ E [(q n+1 ( (q n+1 σ E,σ )+ f(u n+1 This property will be used in the following. ) (qn+1,σ ) f(u n+1,σ )) = (q n+1 σ E,σ ) δ n+1,σ f(u). (3.11) In Section (3.2.2) we shall prove the existence (Lemma 3.1) and the uniqueness (Lemma 3.4) of the solution u = (u n+1 ) M,n [0,N ] to (3.5)-(3.10). We may then define the approximate solution to (3.1)-(3.3) associated to a adapted pointed polygonal finite volume space-time discretization of D of (0, T ) by: Definition 3.1 Let D be a adapted pointed polygonal finite volume space-time discretization of (0, T ) in the sense of Definitions 1.4 and 1.7. The approximate solution of (3.1)-(3.3) associated to the discretization D is defined almost everywhere in (0, T ) by: u D (x, t) = u n+1, x, t (tn, t n+1 ), M, n [[0, N]], (3.12) where (u n+1 ) M,n [0,N ] is the unique solution to (3.5)-(3.10) A numerical example We finally present some numerical results which we obtained by implementing the scheme which was studied above in a prototype code. The domain is the unit square (0, 1) (0, 1). We define two subregions 1 = (0.1, 0.3) (0.4, 0.6) and 2 = (0.7, 0.9) (0.4, 0.6). The initial data is given by 0.5 in \ ( 1 2 ), 1 in 1 and 0 in 2. It is represented on upper left corner of the figure below. The boundary value is the constant 0.5. The function ϕ is defined by ϕ(s) = 0 if s [0, 0.5] and ϕ(s) = 0.2(s 0.5) if s [0.5, 1], so that the diffusion effect only takes place in the areas where the saturation u is greater than.5. The function f is defined by f(s) = s and the field q is defined by q(x, y) = (10(x x 2 )(1 2y), 10(y y 2 )(1 2x)). Hence there is a linear rotating convective transport. We define a coarse mesh of 14 admissible triangles on the unit square, from which we obtain a fine mesh of triangles by refining these 14 triangles uniformly 30 times. This fine mesh is used for the computations. The figure below presents the obtained results at times 0.000, 0.007, and The black points correspond to the value 1, the white ones to the value 0, with a continuous scale of greys between these values. One observes that the initial value 0 is transported, only modified by the numerical diffusion due to the convective upstream weighting, and that, on the contrary, the initial value 1 is rapidly smoothed, due to the effect of the parabolic term which is active on the range [0.5, 1].

39 38 Computed solution at time t = 0 (initial condition), t = 0.007, t = and t = Further analysis Continuous definitions and main convergence result Definition 3.2 (Entropy weak solution) Under Assumption 3.1, a function u is said to be an entropy weak solution to (3.1)-(3.3) if it verifies: u L ( (0, T )), (3.13) ϕ(u) ϕ(u) L 2 (0, T ; H 1 0 ()), (3.14) and u satisfies the following ruzkov entropy inequalities: ψ D + ( [0, T )), κ R, u(x, t) κ ψ t (x, t)+ (f(u(x, t) κ) f(u(x, t) κ)) q(x, t) ψ(x, t) dxdt + u 0 (x) κ ψ(x, 0)dx 0, (0,T ) ϕ(u)(x, t) ϕ(κ) ψ(x, t) (3.15) where one denotes by a b the maximum value between two real values a and b, and by a b their minimum value and where D + ( [0, T )) = {ψ Cc ( R, R + ), ψ(, T ) = 0}.