div(λ(x) u) = f in Ω, (1) u = 0 on Ω, (2) where we denote by Ω = Ω \ Ω the boundary of the domain Ω, under the following assumptions:

Transcription

1 Discretisation of heterogeneous an anisotropic iffusion problems on general nonconforming meshes SUSHI: a scheme using stabilisation an hybri interfaces 1 R. Eymar 2, T. Gallouët 3 an R. Herbin 4 Abstract: A symmetric iscretisation scheme for heterogeneous anisotropic iffusion problems on general meshes is evelope an stuie. The unknowns of this scheme are the values at the centre of the control volumes an at some internal interfaces which may for instance be chosen at the iffusion tensor iscontinuities. The scheme is therefore completely cell-centre if no ege unknown is kept. It is shown to be accurate on several numerical examples. Convergence of the approximate solution to the continuous solution is prove for general (possibly iscontinuous) tensors, general (possibly nonconforming) meshes, an with no regularity assumption on the solution. An error estimate is then euce uner suitable regularity assumptions on the solution. eywors : Heterogeneous anisotropic iffusion, nonconforming gris, finite volume schemes 1 Introuction Anisotropic heterogeneous iffusion problems arise in a wie range of scientific fiels such as hyrogeology, oil reservoir simulation, plasma physics, semiconuctor moelling, biology, etc.. When implementing numerical methos for this kin of problem, one nees to fin an approximation of u, weak solution to the following equation: with bounary conition iv(λ(x) u) = f in, (1) u = 0 on, (2) where we enote by = \ the bounary of the omain, uner the following assumptions: is an open boune connecte polyheral subset of R, N \ {0}, (3) Λ is a measurable function from to M (R), (4) where we enote by M (R) the set of matrices, such that for a.e. x, Λ(x) is symmetric, an such that the set of its eigenvalues is inclue in [λ, λ], with λ an λ R satisfying 0 < λ λ, an f L 2 (). (5) Uner these hypotheses, the weak solution of (1) (2) is the unique function u satisfying: u H0 1(), Λ(x) u(x) v(x)x = f(x)v(x)x v H0 1 (). (6) Usual iscretisation schemes for Problem (6) inclue finite ifference, finite element or finite volume methos. Finite volume methos are actually very popular in oilreservoir engineering, a probable reason being that complex couple physical phenomena may be iscretise on the same gris. The well-known five-point scheme on rectangles (see e.g. [29]) an four-point scheme on triangles [23] are 1 This work was supporte by Groupement MOMAS, CNRS/PACEN 2 Université Paris-Est, France, Robert.Eymar@univ-mlv.fr 3 Université e Provence, France, Thierry.Gallouet@cmi.univ-mrs.fr 4 Université e Provence, France, Raphaele.Herbin@cmi.univ-mrs.fr 1

2 not easily aapte to heterogeneous anisotropic iffusion operators [24]. A scheme with an enlarge stencil, which hanles anisotropy on meshes satisfying an orthogonality property, was propose an analyse in [18]. Another problem that has to be face in several fiels of applications (such as hyrogeology an oil reservoir engineering) is the fact that the iscretisation meshes are impose by engineering an computing consierations; therefore, we have to eal with istorte an possibly nonconforming meshes. A huge literature exists in the engineering setting, so we shall not try to be exhaustive. Let us nevertheless mention the finite volume schemes using the well-known multipoint flux approximation [1, 2, 3]. These schemes involve the reconstruction of the graient in orer to evaluate the fluxes, which is also the case in [13, 28]. Among other approaches let us cite [22], which uses a parametrisation technique. However, even though these schemes perform well in a number of cases, their convergence analysis often seems to remain out of reach, except uner aitional geometrical conitions [13]. More recently, finite volume schemes using interface values have been stuie. In [19] we presente a hybri finite volume (HVF) scheme for any space imension, which involves ege unknowns in aition to the usual cell unknowns, an in [15], a mixe finite volume scheme (MFV) was propose, which involves the fluxes an the values as unknowns. This is also the case for the mimetic finite ifference (MFD) schemes [9, 10], which were introuce previously; in spite of their name, mimetic schemes are very much in the finite volume spirit, since they rely on both a flux balance equation an on the local conservativity of the numerical fluxes, that are probably the two pillars of the finite volume philosophy; but then, finite volume schemes are also often calle finite ifference schemes in the engineering literature because of the finite ifference approximation of the fluxes. In fact, a recent benchmark [25] provie sufficient information to suspect that the methos HFV, MFV an MFD inee coincie at the algebraic level an establishing this is the aim of ongoing work [16]. Let us mention that the Raviart-Thomas mixe finite element metho, which also involves ege unknowns, was generalise to hanle istorte hexaheral meshes [27]. These schemes require the fluxes or ege unknowns as aitional values (or as sole values after hybriisation), an they may be more expensive than cell-centre schemes, especially in the 3D case. In the two-imensional case, we also mention [6], which iscusses a scheme base on vertex reconstructions, an the family of ouble mesh schemes [26, 14, 7]. The generalisation of this type of scheme to 3D is the subject of ongoing work. The scheme that we present here is esigne on very general polygonal, possibly non-convex an nonconforming meshes, with the following two priorities in min: For cost reasons an ata structure issues, we wish to obtain a symmetric scheme which is as close as possible to a cell-centre scheme, that is to a scheme involving one unknown per control volume (or gri cell). For accuracy reasons, we require the local conservativity of the numerical flux to hol at the interfaces between highly heterogeneous meia. In [21], we introuce a cell-centre scheme for the approximation of the Laplace operator on nonconforming gris in the framework of the incompressible Navier-Stokes equations [21] an which may be viewe as a low orer nonconforming Galerkin approximation. The scheme (calle SUCCES in [4]) was also implemente for anisotropic an heterogeneous problems on general meshes, an was shown to be highly competitive for oil reservoir simulation in comparison with other well-known schemes such as the multiple point flux approximation schemes. It is cheaper than the above mentione hybri type schemes (HVF, MFV an MFD) because it is base on cell unknowns only. However, it is not as accurate as the hybri schemes for strongly heterogeneous problems, very likely because of the weaker approximation of the normal fluxes at the heterogeneous interfaces. In the present work, we construct a iscretisation scheme (SUSHI) for any kin of polyheral mesh, which incorporates the best properties of the cell-centre (SUCCES) an hybri (HFV) schemes: unknowns on the eges are 2

3 only introuce when neee, for instance when there is strong meium heterogeneity at these eges. If the set of ege unknowns is empty, then SUSHI reuces to the above mentione cell-centre scheme; if unknowns are associate to all internal eges, then SUSHI is the hybri scheme HFV. The outline of this paper is as follows. In Section 2, we present the guielines which le us in the construction of convergent schemes on general nonconforming meshes. The practical properties of the resulting schemes are shown through numerical examples in Section 3. Then the mathematical analysis of convergence an error estimation are performe in Section 4. This analysis is base on some iscrete functional analytic tools, such as iscrete Sobolev inequalities, which are provie in Section 5. Conclusions an perspectives are iscusse in Section 6. 2 Funamentals for a class of nonconforming schemes Let us first present the esire properties which have le us to the esign of the schemes uner stuy: (P1) The schemes must apply on any type of gri: conforming or nonconforming, 2D an 3D (or more, see for instance the frameworks of kinetic formulations or financial mathematics), consisting of control volumes which are only assume to be polyheral (the bounary of each control volume is a finite union of subsets of hyperplanes). (P2) The matrices of the linear systems generate are expecte to be sparse, symmetric an positive efinite. (P3) We wish to be able to prove the convergence of the family of iscrete solutions to the solution of the continuous problem as the mesh size tens to 0, an of the family of associate graients to the graient of the solution, with no regularity assumption on the solution of the continuous problem, an to erive error estimates when the analytic solution is regular enough. In orer to escribe the schemes we now introuce some notations for the space iscretisation. Definition 2.1 (Space iscretisation) Let be a polyheral open boune connecte subset of R, with N \ {0}, an = \ its bounary. A iscretisation of, enote by D, is efine as the triplet D = (M, E, P), where: 1. M is a finite family of nonempty connecte open isjoint subsets of (the control volumes ) such that =. For any M, let = \ be the bounary of ; let > 0 enote the measure of an let h enote the iameter of. 2. E is a finite family of isjoint subsets of (the eges of the mesh), such that, for all σ E, σ is a nonempty open subset of a hyperplane of R, whose ( 1)-imensional measure is strictly positive. We also assume that, for all M, there exists a subset E of E such that = σ E σ. For any σ E, we enote by M σ = { M, σ E }. We then assume that, for all σ E, either M σ has exactly one element an then σ (the set of these interfaces, calle bounary interfaces, is enote by E ext ) or M σ has exactly two elements (the set of these interfaces, calle interior interfaces, is enote by E int ). For all σ E, we enote by x σ the barycentre of σ. For all M an σ E, we enote by n,σ the unit vector normal to σ outwar to. 3. P is a family of points of inexe by M, enote by P = (x ), such that for all M, x an is assume to be x -star-shape, which means that for all x, the inclusion [x, x] hols. Denoting by the Eucliean istance between x an the hyperplane incluing σ, one assumes that > 0. We then enote by D,σ the cone with vertex x an basis σ. 3

4 Remark 2.1 The above efinition applies to a large variety of meshes. Note that no hypothesis is mae on the convexity of the control volumes; in fact, generalise hexahera, i.e. with faces which may be compose of several planar sub-faces may be use. Often encountere in subsurface flow simulations, such hexahera may have up to 12 faces (resp. 24 faces) if each non planar face is compose of two triangles (resp. four triangles), but only 6 neighbouring control volumes. 2.1 From a hybri finite volume scheme... The iea of the hybri schemes (among them one may inclue the mixe finite elements, the mixe finite volume or the mimetic finite ifference schemes) is to fin an approximation to the solution of (1) (2) by setting up a system of iscrete equations for a family of values ((u ), (u σ ) σ E ) in the control volumes an on the interfaces. The number of unknowns is therefore car(m) + car(e). Following the iea of the finite volume framework, Equation (1) is integrate over each control volume M, which formally gives (assuming sufficient regularity on u an Λ) the following balance equation on the control volume : ( σ ) Λ(x) u(x) n,σ γ(x) = f(x)x. The flux σ Λ(x) u(x) n,σγ(x) is approximate by a function F,σ (u) of the values ((u ), (u σ ) σ E ) at the centres an at the interfaces of the control volumes (in all practical cases, F,σ (u) only epens on u an all (u σ ) σ E ). A iscrete equation corresponing to (1) is then: F,σ (u) = f(x)x M. (7) σ E The values u σ on the interfaces are then introuce so as to allow for a consistent approximation of the normal fluxes in the case of an anisotropic operator an a general, possibly nonconforming mesh. We thus have car(e) supplementary unknowns, an nee car(e) equations to ensure that the problem is well pose. For the bounary faces or eges, these equations are obtaine by writing the iscrete counterpart of the bounary conition (2): u σ = 0 σ E ext. (8) Following the finite volume ieas, we may write the continuity of the iscrete flux for all interior eges, that is to say: F,σ (u) + F L,σ (u) = 0, for σ E int such that M σ = {, L}. (9) We now have car(m) + car(e int ) unknowns an equations. Remark 2.2 In the case Λ(x) = λ(x)i, on meshes satisfying an orthogonality conition as mentione in the introuction of this paper (this conition states the orthogonality between the line joining the centres of two neighbouring control volumes with their common interface, see [17, Definition 9.1 p. 762]), a consistent numerical flux is obtaine using the two-point formula F,σ (u) = λ (u u σ )/, where λ is the average value for λ in. Then, writing (9) for all σ E int such that M σ = {, L}, we obtain u σ as a linear combination of u an u L. Plugging this expression into (7), we get a scheme with car(m) equations an car(m) unknowns (see [17, Section 11.1 pp ] for more etails). In the case of a rectangular (resp. triangular) mesh, this is the well-known five points (resp. four points) scheme with harmonic averages of the iffusion. With a proper choice of the expression F,σ (u), which we shall introuce below, this scheme, first introuce in [19], is quite efficient for the simulation of flui flow in heterogeneous meia (where harmonic averages for Λ are preferre to arithmetic averages [5]) an may be shown to converge. This 4

5 scheme oes have one rawback: since the number of unknowns is the sum of the number of control volumes an of interior interfaces, the resulting scheme is quite expensive (although it is sometimes possible to algebraically eliminate the values at the control volumes, as in the mixe hybri finite element metho, see [8, pp ]). Remark 2.3 Note that in the case of regular conforming simplices (triangles in 2D, tetrahera in 3D), there is an algebraic possibility to express the unknowns (u σ ) σ E as local affine combinations of the values (u ) an therefore to eliminate them [31]. The iea is to remark that the linear system constitute by the equations (7) for all M S, where M S is the set of all simplices sharing the same interior vertex S, an (9) for all the interior eges such that M σ M S, presents as many equations as unknowns u σ, for σ S E. Inee, the number of eges in S 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 or the symmetry of the matrix of this system, an this metho oes not apply to other types of meshes than simplicial meshes. In orer to reuce the computational cost of the scheme, we evelope in [21] an iea which is in fact close to the finite element philosophy since we express the finite volume scheme in a weak form; to this en, let us first efine the sets X D an X D,0 where the iscrete unknowns lie, that is to say: X D = {v = ((v ), (v σ ) σ E ), v R, v σ R}, (10) X D,0 = {v X D such that v σ = 0 σ E ext }. (11) Multiplying, for any v X D,0, Equation (7) by the value v of v on the control volume an summing over M leas to: v F,σ (u) = v f(x)x. Using (9), we get the following iscrete weak formulation: with Fin u X D,0 such that: u, v F = v f(x)x, for all v X D,0, (12) u, v F = F,σ (u)(v v σ ). (13) Note that choosing v X D,0 such that v = 1, v L = 0 for any L M, L an v σ = 0 for any σ E yiels (7). Similarly, choosing v X D,0 such that v = 0 for any M, an v σ = 1 an v τ = 0 for any τ E, τ σ leas to (9). Therefore the hybri finite volume scheme (7) (9) is equivalent to the iscrete weak formulation (12) to a nonconforming finite element scheme... We may then choose to use the weak iscrete form (13) as an approximation of the bilinear form a(, ), but with a space of imension smaller than that of X D,0. This can be achieve 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, (14) 5

6 where (βσ ) is a family of real numbers, with βσ 0 only for some control volumes close to σ E int σ, an such that = 1 an x σ = βσ x σ E int. (15) β σ This ensures that if ϕ is a regular function, then ϕ σ = β σ ϕ(x ) is a consistent approximation of ϕ(x σ ) for σ E int. We recall that the values u σ, σ E ext are set to 0 in orer to respect the bounary conitions (2). Hence the new scheme reas: Fin u X D,0 such that u σ = u, v F = v f(x)x, β σ u σ E int, an for all v X D,0 with v σ = β σ v σ E int. This metho has been shown in [21] to be efficient in the case of a problem where Λ = I (for the approximation of the viscous terms in the Navier-Stokes problem). With an appropriate choice for the expression of the numerical flux, it also yiels conservativity in a certain sense (more on this below), but no longer to the classical (in the finite volume framework) equation (9): inee, since the egrees of freeom on the eges are no longer present, one may not use v σ = 1 to recover (9). Note also that taking v = 1 oes not yiel (7). This scheme has been implemente for the iscretisation of the iffusive term in the incompressible Navier Stokes equations on general two- or three-imensional gris, an gives excellent results [11, 12]. Unfortunately, because of poor approximation of the local flux at strongly heterogeneous interfaces, this approach is not sufficient to provie accurate results for some types of flows in heterogeneous meia, as we shall show in Section 3. This is especially true when using coarse meshes, as is often the case in inustrial problems to an optimal compromise? Therefore we now propose a scheme which has the avantage of both techniques: we shall use equation (13) an keep the unknowns u σ on the eges which require them, for instance those where the matrix Λ is iscontinuous: hence (9) will hol for all eges associate to these unknowns; for all other interfaces, we shall impose the values of u using (14), an therefore eliminate these unknowns. Let us ecompose the set E int of interfaces into two nonintersecting subsets, that is: E int = B H, H = E int \ B. The interface unknowns associate with B will be compute by using the barycentric formula (14). Remark 2.4 Note that, although the accuracy of the scheme is increase in practice when the points where the matrix Λ is iscontinuous are locate within the set σ H σ, such a property is not neee in the mathematical stuy of the scheme. Let us introuce the space X D,B X D,0 efine by: X D,B = {v X D such that v σ = 0 for all σ E ext an v σ satisfying (14) for all σ B}. (17) The composite scheme which we consier in this work reas: (16) { Fin u XD,B such that: u, v F = v f(x)x, for all v X D,B. (18) We therefore obtain a symmetric scheme with car(m) + car(h) equations an unknowns. It is thus less expensive while it remains accurate (for the choice of numerical flux given below) even in the case of strong heterogeneity (see section 3). 6

7 Note that with the present scheme, (9) hols for all σ H, but not generally for any σ B. However, fluxes between pairs of control volumes can nevertheless be ientifie. Inee, we may write u, v F = F,σ (u)(v v σ ) + F,σ (u)βσ L (v v L ), an therefore: where an u, v F = H H B L M F,σ (u)(v v σ ) + 1 F,L (u)(v v L ), 2 (,L) N D N D = {(, L) M 2, σ E B, β L σ 0 or σ E L B, β σ 0}, F,L (u) = B F,σ (u)β L σ σ E L B F L,σ (u)β σ. Note that, if (, L) N D, then (L, ) N D an F,L (u) = F L, (u); furthermore, F,L (u) 0 implies (, L) N D, an the scheme s stencil is etermine by the set {L M such that (, L) N D }. Then, taking v = 1 an all other egrees of freeom of v X D,B equal to 0, (18) yiels F,σ (u) + F,L(u) = f(x)x, H L M (,L) N D which shows the finite volume philosophy of the scheme. Remark 2.5 (Other bounary conitions) In the case of Neumann or Robin bounary conitions, the iscrete space X D,B is moifie to inclue the unknowns associate to the corresponing eges, an the resulting iscrete weak formulation is then straightforwar. Remark 2.6 (Extension of the scheme) For consistency reasons, it is preferable that the coefficients βσ associate with σ B be nonzero for points x that lie in the same regularity zone of the solution as x σ (that is with a zone with no iffusion tensor iscontinuity). This is not always easy: inee, in the tilte barrier example escribe in Section 3.3 below, the barrier contains only one layer of gri cells, so that, for an internal interface of this layer, it is ifficult to use points x L that are locate in the same iffusion regularity zone with respect to x. There is, however, no aitional ifficulty to replace (14) in the efinition of (17) by β σ + σ H u σ = β σ σ β σ u + σ H = 1 an x σ = β σ σ u σ σ B, (19) β σ x + σ H β σ σ x σ σ B. This trick solves the consistency issue without switching the ege to the hybri set H, while all the mathematical properties shown below still hol. (20) 7

8 2.4 Construction of the fluxes using a iscrete graient For the efinition of the schemes to be complete, there now remains to explain how we fin a convenient expression for F,σ (u) with respect to the iscrete unknowns. An iea that has been use in several of the schemes referre to in the Introuction is to look for a consistent expression of the flux by using aequate linear combinations of the unknowns; however, referring to the beginning of Section 2, such a reconstruction oes not in general lea to the esire properties (P2) (symmetric efinite positive matrices) an (P3) (convergence). Our iea here is ifferent: it is base on the ientification of the numerical fluxes F,σ (u) through the mesh-epenent bilinear form, F efine in (13), using the expression of a iscrete graient. Inee let us assume that, for all u X D, we have constructe a iscrete graient D u, we then seek a family (F,σ (u)) such that u, v F = F,σ (u)(v v σ ) = D u(x) Λ(x) D v(x)x u, v X D. (21) σ E Remark 2.7 (On the construction of the iscrete fluxes) Note that it is always possible to euce an expression for F,σ (u) satisfying (21), uner the sufficient conition that, for all M an a.e. x, D u(x) is expresse as a linear combination of (u σ u ) σ E, the coefficients of which are measurable boune functions of x. This property is ensure in the construction of D u(x) given below. We prove in Section 4 below that the esire properties (P2) an (P3) hol if the iscrete graient satisfies the following properties: 1. (Weak compactness) For a sequence of space iscretisations of with mesh size tening to 0, if the sequence of associate gri functions is boune in some sense, then their iscrete graient converges at least weakly in L 2 () to the graient of an element of H 1 0 (); 2. (Consistency) If ϕ is a regular function from to R, the iscrete graient of the piece-wise function efine by taking the value ϕ(x ) on each control volume an ϕ(x σ ) on each ege σ is a consistent approximation of the graient of ϕ. Let us first efine: u = 1 (u σ u )n,σ M, u X D, (22) where n,σ is the outwar to normal unit vector, an are the usual measures (volumes, areas, or lengths) of an σ. The consistency of formula (22) stems from the following geometrical relation: n,σ (x σ x ) t = I M, (23) where (x σ x ) t is the transpose of x σ x R, an I is the ientity matrix. Inee, for any linear function efine on by ψ(x) = G x with G R, assuming that u σ = ψ(x σ ) an u = ψ(x ), we get u σ u = (x σ x ) t G = (x σ x ) t ψ, hence (22) leas to u = ψ. Since the coefficient of u in (22) is in fact equal to zero, a re-construction of the iscrete graient D u solely base on (22) cannot lea to a efinite iscrete bilinear form in the general case. Hence, we now introuce a stabilise graient:,σ u = u + R,σ u n,σ, (24) with R,σ u = (u σ u u (x σ x )), (25) 8

9 (recall that is the space imension an is the Eucliean istance between x an σ). We may then efine D u as the piece-wise constant function equal to,σ u a.e. in the cone D,σ with vertex x an basis σ: D u(x) =,σ u for a.e. x D,σ. (26) Note that, from the efinition (25), thanks to (23) an to the efinition (22), we get that R,σ u n,σ = 0 M. (27) We prove in Lemmata 4.2 an 4.3 below that the iscrete graient efine by (22)-(26) inee satisfies the above state weak compactness an consistency properties. In orer to ientify the numerical fluxes F,σ (u) through Relation (21), we put the iscrete graient in the form with Thus, with, y σσ = n,σ + σ n,σ,σ u = (u σ u )y σσ, σ E D u(x) Λ(x) D v(x)x = A σσ = ( 1 ) n,σ (x σ x ) n,σ if σ = σ σ n,σ (x σ x )n,σ otherwise. σ E y σ σ Λ,σ y σ σ (28) A σσ (u σ u )(v σ v ) u, v X D, (29) σ E an Λ,σ = D,σ Λ(x)x. (30) Then we get that the local matrices (A σσ ) σσ E are symmetric an positive, an the ientification of the numerical fluxes using (21) leas to the expression: F,σ (u) = σ E A σσ (u u σ ). (31) Remark 2.8 (Link with the MFD metho) The above technique yiels an explicit construction of a particular MFD metho. Inee, if one chooses x as the centre of mass of, the matrix A efine by (30) is an aequate choice for the matrix W E which is a parameter in the general formulation of the family of MFD methos as propose in [10]. The avantages of the specific matrix A are that: on particular meshes, taking a natural choice for x (for instance the circumcenter for a triangular mesh in the case of a 2D isotropic problem), it egenerates to a iagonal matrix (see Lemma 2.1 below); it is linke to an explicit formulation of a consistent graient, which is use to efine the iscrete bilinear form (29). Note however that the SUSHI scheme efine by (18) is not the MFD metho of [9, 10]; the main reason is that accoring to the choice B, the SUSHI scheme may be either a completely cell-centre scheme, or a partly or fully hybri scheme, while the MFD metho is a pure hybri scheme. Note 9

10 also that in SUSHI, one may take any point in cell for x, while the MFD schemes [9, 10] are constructe with the centre of mass (however, this choice might be generalise). Note that the proceure which we escribe in Section 2.2 to write a cell-centre scheme coul be applie to any mimetic scheme (or low orer mixe finite element scheme) to yiel a centre scheme. However, further investigations are neee to etermine uner what conitions the present convergence analysis extens to mimetic schemes, an conversely, whether the mimetic analysis applies to the SUSHI scheme (ongoing work, [16]). The fluxes efine by (22)-(31) satisfy certain properties which are etaile in Lemma 4.4, an which allow us to prove the convergence of the scheme, as is shown in Theorem 4.1. Note that it seems ifficult to euce such properties from fluxes obtaine by using natural expansions of regular functions. Note also that both Lemma 4.4 an Theorem 4.1 hol for general heterogeneous, anisotropic an possibly iscontinuous fiels Λ, for which the solution u of (6) is not in general more regular than u H 1 0 (). In the case where Λ an u are regular enough, the local flux consistency satisfie by (31) is use in orer to obtain an error estimate, see Theorem 4.2. The coefficient may be replace by any positive real number without any change in the proof of convergence; in fact, for certain problems it can be interesting to use another coefficient, as escribe in [20] for the so calle SUSHI-P scheme (P for parametric, meaning that the user may choose the stabilisation coefficient as well as the set of eges B). The choice is however natural in the sense that with this value, if B =, the scheme boils own in two imensions to the well-known harmonic averaging five points scheme on rectangles an a four-point scheme on triangles; more generally, in any space imension, even if B an taking the most natural value for u σ if σ B, the resulting flux is a two-point flux on meshes that satisfy the superamissibility conition (32), not necessarily with a harmonic averaging of Λ in the case σ B; this is proven in the next lemma. Note that this superamissibility conition is also satisfie by rectangular parallelepipes in three imensions but unfortunately not by tetrahera. Lemma 2.1 (Superamissible mesh an two-point flux) Let D = (M, E, P) be a iscretisation of in the sense of Definition 2.1, satisfying the following superamissibility conition: n,σ = x σ x M, σ E. (32) Let us furthermore assume that Λ(x) = λ(x)i, where λ is a piece-wise constant function from to R, which is equal to a constant λ in each M; then, the inner prouct efine by (21)-(25) reas: u, v F = λ (u u σ )(v v σ ). Moreover, choosing thanks to (32), x σ = ( x L + L,σ x )/( + L,σ ) for σ E int with M σ = {, L} in (15), the scheme (18) is the following two-point flux scheme: F,σ = f(x) x, (33) F,σ = λ λ L ( + L,σ ) λ L,σ + λ L F,σ = λ + L,σ λ L + L,σ (u u L ) if σ E int H, M σ = {, L}, (34) + L,σ + L,σ (u u L ) if σ E int B, M σ = {, L}, (35) F,σ = λ u if σ E ext E, (36) 10

11 Proof. Let us compute u, v F uner the assumptions of Lemma 2.1. From (21) an thanks to (27) we get: u, v D = λ D u(x) D v(x)x = λ u v + R,σ u R,σ v. Now from the efinition (22) an thanks to the assumption (32), the iscrete graient given by (22) may be written as follows: From (23), we get Therefore, we get that v = 1 (v σ v )(x σ x ) M, v X D, (x σ x ) R,σ u R,σ v = (x σ x ) t = I. (u σ u )(v σ v ) u v, which in turn yiels that u, v D = λ (u σ u )(v σ v ). Hence the matrix A only contains the terms F,σ (u) = λ (u u σ ). on the iagonal, an the flux F,σ (u) is given by Then the scheme (18) can be written as a classical cell-centre finite volume scheme, with two-point fluxes F,L (u) = F,L (u) for any σ E int with M σ = {, L}. Inee, in the case σ / B, the above expression of F,σ (u) allows us to get the following expression of u σ from (9): u σ = This yiels the harmonic averaging two-point flux F,L (u) = λ u + λ L L,σ u L λ + λ. L L,σ λ λ L L,σ λ + λ L L,σ (u u L ). In the case σ B, the two-point barycentric formula u σ = ( u L + L,σ u )/( + L,σ ) together with (18) leas to the resulting two-point flux F,L (u) = λ + L,σ λ L + L,σ + L,σ (u u L ). 11

12 3 Numerical results We present some numerical results obtaine with various choices of B in the scheme (18), (13) with the flux (31), which we synthesise here for the sake of clarity: Fin u X D,B (that is (u ), (u σ ) σ H ), such that: F,σ (u)(v v σ ) = v f(x)x, for all v X D,B, with F,σ (u) = (37) (u σ u ) M, σ E. σ E A σσ where the matrices A σσ are efine by (30)-(28). In the following, we shall use the choices B = (HFV), B = E int or B the set of eges which are locate on the iffusion tensor iscontinuity interfaces; this latter choice is reporte as SUSHI-NP (for non parametric) in [20], in contrast with SUSHI-P (for parametric) where the choice of the set B may be ifferent, along with the value of the stabilisation coefficient in (25). 3.1 Implementation Let us first escribe an implementation aspects of the scheme. The unknowns, i.e. the values u, for M an the values u σ, σ E int H, are orere as (u i ) i=1,...,n. The N N matrix an the N 1 right-han-sie of the linear system resulting from (18) are compute thanks to a loop over the control volumes M an to an inner loop on each ege σ E. Let us etail the matrix computation loop. 1. All store matrix coefficients are initially set to The expression F,σ (u) is written in the form F,σ (u) = i=1,...,n a(i),σ u i, where the nonzero coefficients (a (i),σ ) i=1,...,n are only locally compute (they are not store for all an σ). These coefficients are obtaine after the elimination of all (u σ ) σ E B in (31): F,σ (u) = σ E H A σσ (u u σ ) + σ E B A σσ L M β L σ (u u L ). 3. The line of the matrix corresponing to the unknown u is incremente at the column j with the coefficient a (j),σ. 4. If σ B with v σ = L M βl σ v L for any v X D,B, the line of the matrix corresponing to each L M such that β L σ 0 is incremente at the column j with the coefficient β L σ a (j),σ. 5. If σ E int H, the line of the matrix corresponing to the ege σ is incremente at the column j with the coefficient a (j),σ. This proceure is ientical in the cases B = (HFV), B = an B = E int. However, in the case where B = (HFV), one may eliminate the unknowns u with respect to the unknowns u σ, as in the hybri implementation of the mixe finite element metho. 3.2 Orer of convergence We consier here the numerical resolution of Equation (1) supplemente by the homogeneous Dirichlet bounary conition (2); the right-han sie is chosen so as to obtain an exact solution to the problem 12

13 an easily compute the error between the exact an approximate solutions. We consier Problem (1)-(2) with a constant matrix Λ: ( ) Λ =, (38) an choose f : R such that the exact solution to Problem (1) (2) is ū efine 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-centre scheme, there are no ege-unknowns. Let us first consier conforming meshes, such as the triangular meshes which are epicte on Figure 1, an uniform square meshes. Figure 1: Regular conforming coarse an fine triangular gris For both B = (pure hybri scheme: HFV) an B = E int (cell-centre scheme), the orer of convergence is close to 2 for the unknown u an 1 for its graient. Of course, the hybri scheme is almost three times more costly in terms of number of unknowns than the cell-centre 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-centre scheme than for the hybri scheme. Hence the number of unknowns is probably not a sufficient criterion for assessing the cost of the scheme. Results were also obtaine in the case of uniform square or rectangular meshes. They show a better rate of convergence of the graient (orer 2 in the case of H = E int an 1.5 in the case B = E int ), even though the rate of convergence of the approximate solution remains unchange an close to 2. We then use a rectangular nonconforming mesh, obtaine by cutting vertically the omain into two parts an using a rectangular gri of 3n 2n (resp. 5n 2n) on the first (resp. secon sie), where n is the number of the mesh, n = 1,..., 7. Again, the orer of convergence which we obtain is 2 for u an aroun 1.8 for the graient. We give in Table 1 below the errors obtaine in the iscrete L 2 norm for u an u for a nonconforming mesh an (in terms of number of unknowns) an for the rectangular 4 6 an 4 10 conforming rectangular meshes, for both the hybri an cell-centre schemes. We show in Figure 2 the solutions for the corresponing gris (which look much the same for the two schemes). Further etaile results on several problems an conforming, nonconforming an istorte meshes may be foun in [20]. 3.3 The case of a highly heterogeneous tilte barrier We now turn to the heterogeneous case. The omain =]0, 1[ ]0, 1[ is compose of 3 sub-omains, which are epicte in Figure 3: 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}, an δ = 0.2 is the slope of the rain (see Figure 3). Dirichlet bounary conitions are impose by setting the bounary values to those of the analytical solution given by u(x, y) = ϕ 1 (x, y) on 1 3 an u(x, y) = ϕ 1 (x, y)/10 2 on 2. 13

14 NU NM ɛ(u) ɛ( u) n B = B = E int B = B = E int B = B = E int B = B = E int C E E E E-02 NC E E E E-02 C E E E E-02 Table 1: Error for the nonconforming rectangular mesh, pure hybri scheme (B = ) an centre (B = E int ) schemes. For both schemes NU is the number of unknowns in the resulting linear system, NM is the number of nonzero terms in the matrix, ɛ(u) is the iscrete L 2 norm of the error of the solution an ɛ( u) is the iscrete L 2 norm of the error in the graient. C1 an C2 are the two conforming meshes represente on the left an the right in Figure 2, an NC is the nonconforming one represente in the mile. Figure 2: The approximate solution for conforming an nonconforming meshes. 8 6 mesh, centre: nonconforming 4 6, 4 10 mesh, right: conforming Left: conforming The permeability tensor Λ is heterogeneous an isotropic, given by Λ(x) = λ(x)i, with λ(x) = 1 for a.e. x 1 3 an λ(x) = 10 2 for a.e. x 2. Note that the isolines of the exact solution are parallel to the bounaries of the sub-omain, an that the tangential component of the graient is 0. We use the meshes epicte in Figure 3. Mesh 3 (containing control volumes) is obtaine from Mesh 1 by the aition of two layers of very thin control volumes aroun each of the two lines of iscontinuity of Λ: because of the very low thickness of these layers, equal to 1/10000, the picture representing Mesh 3 is not ifferent from that of Mesh Figure 3: Domain an meshes use for the tilte barrier test: mesh 1 (10 21 centre), mesh 2 ( right) We get the following results for the approximations of the four fluxes at the bounary. 14

15 nb. unknowns matrix size x = 0 x = 1 y = 0 y = 1 analytical B = E int mesh mesh mesh SUSHI-NP mesh mesh HFV mesh mesh Note that the values of the numerical solution given by the pure hybri (HFV) an composite (SUSHI- NP) schemes are equal to those of the analytical solution (this hols uner the only conition that the interfaces locate on the lines ϕ i (x, y) = 0, i = 1, 2, are not inclue in B, an that, for all σ B, all M with βσ 0 are inclue in the same sub-omain i ). Note that Mesh 3, which leas to acceptable results for the computation of the fluxes, is not well suite for such a couple problem, because of too small control volume measures. Hence SUSHI on Mesh 1 appears to be the most suitable metho for this problem. A satisfying natural choice (SUSHI-NP in the above results) is thus to match H with the iscontinuities of Λ. It is sometimes interesting to choose another set B. This is for instance the case for the numerical locking problem for which the choice B = is best even though the iffusion tensor is homogeneous [20]. It is also sometimes interesting to replace the stabilisation coefficient in (25) by some other coefficient α > 0. This is the case for instance for very istorte meshes or singular problems, in orer to maintain the positivity of the unknown. The coefficient α is taken to be greater than. The approximate solution remains positive, but the L 2 norm of the error is generally larger. We refer to [20] for such experiments. 4 Convergence of the scheme Let us first introuce some notations relate to the mesh. Let D = (M, E, P) be a iscretisation of in the sense of Definition 2.1. The size of the iscretisation D is efine by: h D = sup{h, M}, an the regularity of the mesh by: ( θ D = max max, σ E int,,l M σ L,σ For a given set B E int an for a given family (β σ ) measure of the resulting regularity by ( θ D,B = max θ D, max, B max h, σ E int ). (39) satisfying property (15), we introuce a ) L M βl σ x L x σ 2. (40) Remark 4.1 Note that, for any mesh, it is easy to choose the family (β σ ) h 2 σ E int so that θ D,B remains small. It suffices to express x σ as the barycentre of + 1 points x L (which is always possible), for L sufficiently close to, so that x L x σ is close to h when βσ 0. Note also that in fact, it woul be sufficient to have h η with η > 1 instea of h2 in (40) thus allowing the use of farther points. 15

16 Remark that, thanks to the assumption that is x -star-shape, the following property hols: = M. (41) The space X D efine in (10) is equippe with the following semi-norm: v X D, v 2 X = (v σ v ) 2, (42) which is a norm on the spaces X D,0 an X D,B respectively efine by (11) an (17). Let H M () L 2 () be the set of piece-wise constant functions on the control volumes of the mesh M. We then enote, for all v H M () an for all σ E int with M σ = {, L}, D σ v = v v L an σ = + L,σ, an for all σ E ext with M σ = {}, we enote D σ v = v an σ =. We then efine the following norm: v H M (), v 1,2,M = ( ) Dσ v 2 = (D σv) 2. (43) σ (Note that this norm is also efine by (74) in Lemma 5.2, setting p = 2). For all v X D, we enote by Π M v H M () the piece-wise function from to R efine 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 leas to the relation σ σ E (v v L ) 2 σ (v v σ ) 2 + (v σ v L ) 2 L,σ v X D, Π M v 2 1,2,M v 2 X v X D,0. (44) For all ϕ C(, R), we enote by P D ϕ the element of X D efine by ((ϕ(x )), (ϕ(x σ )) σ E ), by P D,B ϕ the element v X D,B such that v = ϕ(x ) for all M, v σ = 0 for all σ E ext, v σ = β σ ϕ(x ) for all σ B an v σ = ϕ(x σ ) for all σ H. We enote by P M ϕ H M () the function such that P M ϕ(x) = ϕ(x ) for a.e. x, for all M (we then have P M ϕ = Π M P D ϕ = Π M P D,B ϕ). The following lemma provies an equivalence property between the L 2 -norm of the iscrete graient, efine by (22)-(26) an the norm X. Lemma 4.1 Let D be a iscretisation of in the sense of Definition 2.1, an let θ θ D be given (where θ D is efine by (39)). Then there exists C 1 > 0 an C 2 > 0 only epening on θ an such that: C 1 u X D u L 2 () C 2 u X u X D, (45) where D is efine by (22)-(26). Proof. By efinition, Therefore, using property (27), D u 2 L 2 () = D u 2 L 2 () =,σ u 2. u 2 + (R,σ u) 2. (46) 16

17 Let us now notice that the following inequality hols: (a b) 2 λ 1 + λ a2 λb 2 a, b R, λ > 1. (47) We apply this inequality to (R,σ u) 2 for some λ > 0 an obtain This leas to (R,σ u) 2 λ 1 + λ Choosing λ = 1 θ 2, we get that (R,σ u) 2 λ 1 + λ ( ) uσ u 2 ( ) λ u 2 xσ x 2. (48) D u 2 (L 2 ()) ( ) uσ u 2 λ u 2 θ 2. λ 1 + λ u 2 X, which shows the left inequality of (45). Let us now prove the right inequality. On one han, using the efinition (22) of u an (41), the Cauchy Schwarz inequality leas to u (u σ u ) 2 = (u σ u ) 2. (49) On the other han, by efinition (25), an thanks to the efinition of the regularity of the mesh (39), we have ( (R,σ u) 2 2 ( u σ u ) 2 + u 2 x ) ( σ x 2 2 ( u ) σ u ) 2 + θ 2 u 2. (50) From (46), (49) an (50), we conclue that the right inequality of (45) hols. We may now state a weak compactness result for the iscrete graient. Lemma 4.2 (Weak iscrete H 1 compactness) Let F be a family of iscretisations in the sense of Definition 2.1 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, there exists C > 0 with u D X C for all D F, there exists u L 2 () with lim Π Mu D u L h D 0 2 () = 0. Then, u H 1 0 () an Du D weakly converge in L 2 () to u as h D 0, where the operator D is efine by (22)-(26). Proof. Let us prolong Π M u D an D u D by 0 outsie of. Thanks to Lemma 4.1, up to a subsequence, there exists some function G L 2 (R ) such that D u D weakly converges in L 2 (R ) to G as h D 0. Let us show that G = u. Let ψ Cc (R ) be given. Let us consier the term T1 D efine by T1 D = D u D (x) ψ(x)x. R 17

18 We get that T D 1 = T D 2 + T D 3, with an T D 2 = We compare T D 2 with T D 4 with We get that (u σ u )n,σ ψ, with ψ = 1 T D 3 = efine by T D 4 (T D 2 T D 4 ) 2 = R,σ u n,σ ψ(x)x. D,σ (u σ u )n,σ ψ σ, ψ σ = 1 ψ(x)γ(x). σ (u σ u ) 2 ψ(x)x, ψ ψ σ 2, which leas to lim (T 2 D T 4 D) = 0. h D 0 Since T4 D = u n,σ ψ σ = Π M u D (x)ivψ(x)x, σ E R we get that lim T 4 D h D 0 (27) hols, we have: = R u(x)ivψ(x)x. Let us now turn to the stuy of T D 3 T D 3 = R,σ u n,σ (ψ(x) ψ )x. D,σ. Noting again that Since ψ is a regular function, there exists C ψ only epening on ψ such that D,σ (ψ(x) ψ )x C ψ h D. From (50) an the Cauchy-Schwarz inequality, we thus get: lim T 3 D = 0. h D 0 This proves that the function G L 2 (R ) is a.e. equal to u in R. Since u = 0 outsie of, we get that u H0 1(), an the uniqueness of the limit implies that the whole family Du D weakly converges in L 2 (R ) to u as h D 0. Note that the proof that u H0 1 () also results from (44), which allows us to apply Lemma 5.7 of the Appenix in the particular case p = 2. Let us also remark that several iscrete graients coul be chosen, which satisfy the weak compactness property (see for instance the proof of Lemma 5.7). However, we emphasise that the choice of the specific graient (22) also stems from coercivity an consistency issues. Let us now state the iscrete graient consistency property. Lemma 4.3 (Discrete graient consistency) Let D be a iscretisation of in the sense of Definition 2.1, an let θ θ D be given. Then, for any function ϕ C 2 (), there exists C 3 only epening on, θ an ϕ such that: D P D ϕ ϕ (L ()) C 3 h D, (51) where D is efine by (22)-(26). 18

19 Proof. From efinitions (26) an (24) we get From (22), 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 = 1 (ϕ(x σ ) ϕ(x ))n,σ ( ϕ(x ) (x σ x ) + h 2 ) ρ,σ n,σ, where ρ,σ C ϕ with C ϕ only epening on ϕ. Thanks to (23) an to the regularity of the mesh, we get P D ϕ ϕ(x ) 1 h 2 ρ,σ h C ϕ θ. From this last inequality, using Definition 25, we get R,σ P D ϕ = ϕ(x σ ) ϕ(x ) P D ϕ (x σ x ) ( h 2 ρ,σ + h 2 C ϕ θ ),σ θ(h C ϕ + h C ϕ θ), which conclues the proof. We now give the abstract properties of the iscrete fluxes, which are necessary to prove the convergence of the general scheme (18), (13), an then prove that the fluxes that we constructe in Section 2.4 inee satisfy these properties. Definition 4.1 (Continuous, coercive, consistent an symmetric families of fluxes) Let F be a family of iscretisations in the sense of efinition 2.1. For D = (M, E, P) F, M an σ E, we enote by F,σ D a linear mapping from X D to R, an we enote by Φ = ((F,σ D ) ) D F. We consier the bilinear form efine by u, v F = σ E F D,σ(u)(v v σ ) (u, v) X 2 D. (52) The family of numerical fluxes Φ is sai 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. (53) The family of numerical fluxes Φ is sai 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. (54) The family of numerical fluxes Φ is sai to be consistent (with Problem (1) (2)) 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 that u H 1 0 ()), lim Π Mu D u L h D 0 2 () = 0 (recall that, from Lemma 5.7, we get 19

20 then lim D, P D ϕ F = Λ(x) ϕ(x) u(x)x h D 0 ϕ Cc (). (55) Finally the family of numerical fluxes Φ is sai to be symmetric if u, v F = v, u F (u, v) X 2 D, D = (M, E, P) F. We now show that the family of fluxes efine by (28)-(31) satisfies the efinition of a consistent, coercive an symmetric family of fluxes. Recall that the SUSHI scheme (37) is stuie numerically in Section 3 with this choice for the family of fluxes. Lemma 4.4 (Flux properties) Let F be a family of iscretisations in the sense of Definition 2.1. We assume that there exists θ > 0 with θ D θ D = (M, E, P) F, (56) where θ D is efine by (39). Let Φ = ((F D,σ ) ) D F be the family of fluxes efine by (28)-(31). Then, the family Φ is a continuous, coercive, consistent an symmetric family of numerical fluxes in the sense of Definition 4.1. Proof. Since the family of fluxes is efine by (28)-(31), it satisfies (21), an therefore we have: u, v F = D u(x) Λ(x) D v(x)x u, v X D. Hence the property u, v F = v, u F hols. The continuity an coercivity of the family Φ result from Lemma 4.1 an the properties of Λ, which give: u, v F λ D u L 2 () D v L 2 () an u, u F λ D u 2 L 2 () for any u, v X D. The consistency results from the weak an strong convergence properties in Lemmas 4.2 an 4.3, which give D u D u weakly in L 2 () an D P D ϕ ϕ in L 2 () as the mesh size tens to 0. Theorem 4.1 (Convergence) Let F be a family of iscretisations in the sense of Definition 2.1, for any D F, let B E int an (βσ ) satisfying (15). Assume that there exists θ > 0 such that σ E int θ D,B θ, for all D F, where θ D,B is efine by (40). Let Φ = ((F,σ D ) ) D F be a continuous, coercive an symmetric an consistent family of numerical fluxes in the sense of Definition 4.1. Let (u D ) D F be the family of functions satisfying (18) for all D F. Then Π M u D converges in L 2 () to the unique solution u of (6) as h D 0. Moreover D u D converges to u in L 2 () as h D 0. Proof. Letting v = u D in (18) an applying the Cauchy-Schwarz inequality yiels u D, u D F = f(x)π M u D (x)x f L 2 () Π M u D L 2 (). We apply the Sobolev inequality (77) with p = 2, which gives in this case Π M u D L 2 () C 4 Π M u 1,2,M. Using (44) an the consistency of the family Φ of fluxes, we then have α Π M u D 2 X C 4 f L 2 () u D X. σ E This leas to the inequality u D 1,2,M u D X C 4 α f L 2 (). (57) 20

21 Thanks to Lemma 5.7, we get the existence of u H0 1 (), an of a subfamily extracte from F, such that Π M u D u L 2 () tens to 0 as h D 0. For a given ϕ Cc (), let us take v = P D,B ϕ in (18) (recall that P D,B ϕ X D,B ). We get u D, P D,B ϕ F = f(x)p M ϕ(x)x. Let us remark that, thanks to the continuity of the family Φ of fluxes, we have u D, P D,B ϕ P D ϕ F M C 13 α f L 2 () P D,B ϕ P D ϕ X. Thanks to (15) an (40), we get the existence of C ϕ only epening on ϕ (through its secon orer partial erivatives) such that, for all M an all σ B E, βσ L ϕ(x L ) ϕ(x σ ) L M L M β L σ x L x σ 2 C ϕ θ D,B C ϕ h 2. (58) We can then euce lim P D,Bϕ P D ϕ X = 0. (59) h D 0 Thanks to the F-extracte subfamily properties, we may 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)x. h D 0 Gathering the two results above leas to lim u D, P D,B ϕ F = h D 0 which conclues the proof of the following equality Λ(x) ϕ(x) u(x)x = Λ(x) ϕ(x) u(x)x, f(x)ϕ(x)x. Therefore, u is the unique solution of (6), an we get that the whole family (u D ) D F converges to u as h D 0. Let us now prove the secon part of the theorem. Let ϕ C c () be given (this function is evote to approximate u in H0 1()). Cauchy-Schwarz inequality, we have D u D (x) u(x) 2 x 3 (T5 D + T6 D + T 7 ), Thanks to the with T D 5 = Du D (x) D P D ϕ(x) 2 x, T D 6 = DP D ϕ(x) ϕ(x) 2 x, an T 7 = ϕ(x) u(x) 2 x. Thanks to Lemma 4.3, we have lim hd 0 T D 6 = 0. Thanks to Lemma 4.1 an to the coercivity of the family of fluxes, there exists C 5 such that with C 5 = C 2 2 α. Taking v = u D P D ϕ, we have D v 2 L 2 () C 2 2 v 2 X C 5 v, v F v X D, T D 5 C 5 ( u D, u D F 2 u D, P D ϕ F + P D ϕ, P D ϕ F ). 21