A mixed finite volume scheme for anisotropic diffusion problems on any grid

HAL Id: hal-00005565

https://hal.archives-ouvertes.fr/hal-00005565v3

Submitted on 3 Apr 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A mixed finite volume scheme for anisotropic diffusion problems on any grid

J. Droniou, Robert Eymard

To cite this version:

J. Droniou, Robert Eymard. A mixed finite volume scheme for anisotropic diffu- sion problems on any grid. Numerische Mathematik, Springer Verlag, 2006, 105, http://rd.springer.com/article/10.1007/s00211-006-0034-1. �10.1007/s00211-006-0034-1�. �hal- 00005565v3�

ccsd-00005565, version 3 - 3 Apr 2006

A mixed finite volume scheme for anisotropic diffusion problems on any grid

J´erˆome Droniou¹ and Robert Eymard ²

27/03/2006 Abstract

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. The approximate solution is shown to converge to the continuous one as the size of the mesh tends to 0, and an error estimate is given. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids and for various diffusion matrices.

Keywords. Finite volume scheme, unstructured grids, irregular grids, anisotropic heteroge- neous diffusion problems.

1 Introduction

The computation of an approximate solution for equations involving a second order elliptic operator is needed in so many physical and engineering areas, where the efficiency of some discretization methods, such as finite difference, finite element or finite volume methods, has been proven. The use of finite volume methods is particularly popular in the oil engineering field, since it allows for coupled physical phenomena in the same grids, for which the conservation of various extensive quantities appears to be a main feature. However, it is more challenging to define convergent finite volume schemes for second-order elliptic operators on refined, distorted or irregular grids, designed for the purpose of another problem.

For example, in the framework of geological basin simulation, the grids are initially fitted on the geological layers boundaries, which is a first reason for the loss of orthogonality. Then, these grids are modified during the simulation, following the compaction of these layers (see [14]), thus leading to irregular grids, as those proposed by [16]. As a consequence, it is no longer possible to compute the fluxes resulting from a finite volume scheme for a second order operator, by a simple two-point difference across each interface between two neighboring control volumes. Such a two-point scheme is consistent only in the case of an isotropic operator, using a grid such that the lines connecting the centers of the control volumes are orthogonal to the edges of the mesh.

The problem of finding a consistent expression using only a small number of points, for the finite volume fluxes in the general case of any grid and any anisotropic second order operator, has led to many works (see [1], [2], [3], [14] and references therein; see also [19]). A recent finite volume scheme has been proposed [11, 12], permitting to obtain a convergence property in the case of an

1D´epartement de Math´ematiques, UMR CNRS 5149, CC 051, Universit´e Montpellier II, Place Eug`ene Batail- lon, 34095 Montpellier cedex 5, France. email: droniou@math.univ-montp2.fr

2Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UMR 8050, Universit´e de Marne-la-Vall´ee, 5 boule- vard Descartes, Champs-sur-Marne, 77454 Marne-la-Vall´ee Cedex 2, France. email:Robert.Eymard@univ-mlv.fr

anisotropic heterogeneous diffusion problem on unstructured grids, which all the same satisfy the above orthogonality condition. In the case where such an orthogonality condition is not satisfied, a classical method is the mixed finite element method which also gives an approximation of the fluxes and of the gradient of the unknown (see [4], [5], [6], [22] for example, among a very large literature). Note that, although the Raviart-Thomas basis is not directly available on control volumes which are not simplices or regular polyhedra, such a basis can be built on more general irregular grids. In [17], such a construction is completed using decomposition into simplices and a local elimination of the unknowns at the internal edges. In [9] and [13], such basis functions are obtained from the resolution of a Neumann elliptic problem in each grid block. However, it has been observed that the use of mixed finite element method could demand high refined grids on some highly heterogeneous and anisotropic cases (see [18] and the numerical results provided in the present paper). Note that an improvement of the mixed finite element scheme is the expanded mixed finite element scheme [7], where different discrete approximations are proposed for the unknown, its gradient and the product of the diffusion matrix by the gradient of the unknown. However, this last scheme seems to present the same restrictions on the meshes as the mixed finite element scheme.

We thus propose in this paper an original finite volume method, called the mixed finite volume method, which can be applicable on any type of grids in any space dimension, with very few restrictions on the shape of the control volumes. The implementation of this scheme is proven to be easy, and no geometric complex shape functions have to be computed. Accurate results are obtained on coarse grids in the case of highly heterogeneous and anisotropic problems on irregular grids. In order to show the mathematical and numerical properties of this scheme, we study here the following problem: find an approximation of ¯u, weak solution to the following problem:

−div(Λ∇u) =¯ f in Ω,

¯

u= 0 on∂Ω, (1)

under the following assumptions:

Ω is an open bounded connected polygonal subset ofR^d, d≥1, (2) Λ : Ω→M_d(R) is a bounded measurable function such that

there exists α0 >0 satisfying Λ(x)ξ·ξ≥α0|ξ|² for a.e. x∈Ω and all ξ∈R^d, (3) (where M_d(R) stands for the space ofd×dreal matrices) and

f ∈L²(Ω). (4)

Thanks to Lax-Milgram theorem, there exists a unique weak solution to (1) in the sense that

¯

u∈H₀¹(Ω) and the equation is satisfied in the sense of distributions on Ω.

The principle of the mixed finite volume scheme, described in Section 2, is the following. We simultaneously look for approximations u_K and v_K in each control volume K of ¯u and ∇u,¯ and find an approximation F_σ at each edge σ of the mesh of R

σΛ(x)∇u(x)¯ ·n_σdγ(x), where n_σ is a unit vector normal toσ. The values F_σ must then satisfy the conservation equation in each control volume, and consistency relations are imposed on uK, vK and Fσ. After having investigated in Section 3 the properties of a space associated with the scheme, we show in Section 4 that it leads to a linear system which has one and only one approximate solution u, v and

F, and we provide the mathematical analysis of its convergence and give an error estimate.

In Section 5, we propose an easy implementation procedure for the scheme, and we use it for the study of some numerical examples. We thus obtain acceptable results on some grids for which it would be complex to use other methods, or to which empirical methods apply but no mathematical results of convergence nor stability have yet been obtained.

2 Definition of the mixed finite volume scheme and main results

We first present the notion of admissible discretization of the domain Ω, which is necessary to give the expression of the mixed finite volume scheme.

Definition 2.1 [Admissible discretization] LetΩbe an open bounded polygonal subset ofR^d (d≥1), and∂Ω = Ω\Ω its boundary. An admissible finite volume discretization of Ω is given byD= (M,E,P), where:

• Mis a finite family of non empty open polygonal convex disjoint subsets ofΩ(the “control volumes”) such that Ω =∪K∈MK.

• E is a finite family of disjoint subsets of Ω (the “edges” of the mesh), such that, for all σ∈ E, there exists an affine hyperplane E of R^dand K ∈ Mwith σ⊂∂K∩E andσ is a non empty open convex subset ofE. We assume that, for all K∈ M, there exists a subset EK of E such that ∂K =∪σ∈EKσ. We also assume that, for all σ ∈ E, either σ ⊂∂Ω or σ=K∩L for some (K, L) ∈ M × M.

• P is a family of points of Ω indexed by M, denoted by P = (x_K)_K∈M and such that, for allK ∈ M, x_K ∈K.

Some examples of admissible meshes in the sense of the above definition are shown in Figures 1 and 2 in Section 5.

Remark 2.1 Though the elements ofEK may not be the real edges of a control volumeK (each σ∈ EK may be only a part of a full edge, see figure 2), we will in the following call “edges of K”

the elements of EK.

Notice that we could also cut each intersection K∩L into more than one edge. This would not change our theoretical results but this would lead, for practical implementation, to artificially enlarge the size of the linear systems to solve, which would decrease the efficiency of the scheme.

Remark 2.2 The whole mathematical study done in this paper applies whatever the choice of the point x_K in each K ∈ M. In particular, we do not impose any orthogonality condition connecting the edges and the points xK. However, the magnitude of the numerical error (and, for some regular or structured types of mesh, its order) does depend on this choice.

We could also extend our definition to non-planar edges, under some curvature condition. In this case, it remains possible to use the mixed finite volume scheme studied in this paper and to prove its convergence.

The following notations are used. The measure of a control volumeK is denoted by m(K); the (d−1)-dimensional measure of an edgeσ is m(σ). In the case whereσ ∈ E is such thatσ=K∩L

for (K, L) ∈ M × M, we denote σ =K|L. For all σ ∈ E, x_σ is the barycenter of σ. Ifσ ∈ EK

thenn_K,σ is the unit normal to σ outward to K. The set of interior (resp. boundary) edges is denoted byEint (resp. Eext), that isEint ={σ∈ E;σ 6⊂∂Ω}(resp. Eext={σ∈ E;σ ⊂∂Ω}). For all K∈ M, we denote byNK the subset of Mof the neighboring control volumes (that is, the L such thatK∩L is an edge of the discretization).

To study the convergence of the scheme, we will need the following two quantities: the size of the discretization

size(D) = sup{diam(K) ; K ∈ M}

and the regularity of the discretization regul(D) = sup

max

diam(K)^d

ρ^d_K ,Card(EK)

; K∈ M

(5) where, for K ∈ M,ρ_K is the supremum of the radius of the balls contained inK. Notice that, for all K∈ M,

diam(K)^d≤regul(D)ρ^d_K ≤ regul(D)

ω_d m(K) (6)

where ω_d is the volume of the unit ball in R^d. Note also that regul(D) does not increase in a local refinement procedure, which will allow the scheme to handle such procedures.

We now define the mixed finite volume scheme. Let D be an admissible discretization of Ω in the sense of Definition 2.1. Denote byH_D the set of real functions on Ω which are constant on each control volumeK ∈ M(if h∈H_D, we leth_K be its value on K).

As said in the introduction, the idea is to consider three sets of unknowns, namely u ∈ H_D which approximates ¯u, v ∈ H_D^d which approximates ∇u¯ and a family of real numbers F = (F_K,σ)_{K∈M,σ∈EK} (we denote byFDthe set of such families) which approximates (R

σΛ(x)∇u(x)¯ · n_K,σdγ(x))_{K∈M,σ∈EK}.

Taking ν = (ν_K)_K∈M a family of nonnegative numbers, we define L_ν(D) as the space of (u,v, F)∈HD×H_D^d × FD such that

vK·(xσ−xK) +vL·(xL−xσ) +νKm(K)FK,σ−νLm(L)FL,σ=uL−uK,

∀K ∈ M, ∀L∈ NK, withσ =K|L, v_K·(x_σ−x_K) +ν_Km(K)F_K,σ =−u_K, ∀K ∈ M, ∀σ∈ EK∩ Eext

(7)

and we define the mixed finite volume scheme as: find (u,v, F)∈L_ν(D) such that

F_K,σ+F_L,σ= 0, ∀σ=K|L∈ Eint, (8) m(K)Λ_Kv_K= X

σ∈EK

F_K,σ(x_σ−x_K), ∀K ∈ M (9) (where Λ_K = _m(K)¹ R

KΛ(x) dx) and

− X

σ∈EK

F_K,σ = Z

K

f(x) dx, ∀K∈ M. (10)

The origin of each of these equations is quite easy to understand. Since u and v stand for approximate values of ¯u and ∇u, equation (7) simply states, if we assume¯ ν_K = 0, that v is

a discrete gradient of u: it is the discrete counterpart of u(x_L)−u(x_K) = u(x_L)−u(x_σ) + u(x_σ)−u(x_K)≈ ∇u(x_L)·(x_L−x_σ) +∇u(x_K)·(x_σ−x_K). This equation is slightly penalized with the fluxes to ensure existence and estimates on the said fluxes (to study the convergence of the scheme, we will assume ν_K > 0; see the theorems below). Notice that the boundary condition ¯u= 0 is contained in the second line of (7). AsF_K,σ stands for an approximate value of R

σΛ∇(x)¯u(x)·nK,σdγ(x), it is natural to ask for the conservation property (8), and the balance (10) simply comes from an integration of (1) on a control volume. Last, the link (9) between Λv and its fluxes is justified by Lemma 6.1 in the appendix, which shows that one can reconstruct a vector from its fluxes through the edges of a control volume.

Our main results on the mixed finite volume scheme are the following. The first one states that there exists a unique solution to the scheme. The second one gives the convergence of this solution to the solution of the continuous problem, as the size of the mesh tends to 0, and the third one provides an error estimate in the case of smooth data.

Theorem 2.1 Let us assume Assumptions (2)-(4). Let D be an admissible discretization of Ω in the sense of Definition 2.1. Let (ν_K)_K∈M be a family of positive real numbers. Then there exists one and only one (u,v, F) solution of ((7),(8),(9),(10)).

Theorem 2.2 Let us assume Assumptions (2)-(4). Let (Dm)m≥1 be admissible discretizations of Ω in the sense of Definition 2.1, such that size(Dm) → 0 as m → ∞ and (regul(Dm))_m≥1 is bounded. Let ν₀ > 0 and β ∈ (2−2d,4−2d) be fixed. For all m ≥ 1, let (u_m,v_m, F_m) be the solution to ((7),(8),(9),(10)) for the discretization Dm, setting νK = ν0diam(K)^β for all K ∈ Mm. Letu¯ be the weak solution to (1).

Then, as m → ∞, v_m → ∇u¯ strongly in L²(Ω)^d and u_m →u¯ weakly in L²(Ω)and strongly in L^q(Ω)for all q <2.

Theorem 2.3 Let us assume Assumptions (2)-(4). Let D be an admissible discretization of Ω in the sense of Definition 2.1, such that size(D) ≤ 1 and regul(D) ≤ θ for some θ > 0. We take ν₀ > 0 and β ∈ (2−2d,4 −2d) and, for all K ∈ M, we let ν_K = ν₀diam(K)^β. Let (u,v, F) be the solution to ((7),(8),(9),(10)). Letu¯ be the weak solution to (1). We assume that Λ∈C¹(Ω;M_d(R)) andu¯∈C²(Ω).

Then there exists C₁ only depending on d, Ω, u,¯ Λ, θ and ν₀ such that

kv− ∇u¯kL²(Ω)^d ≤C₁size(D)¹²min(β+2d−2,4−2d−β) (11) and

ku−u¯kL²(Ω) ≤C₁size(D)¹²min(β+2d−2,4−2d−β) (12) (note that the maximum value of ¹₂min(β+ 2d−2,4−2d−β) is ¹₂, obtained for β= 3−2d).

Remark 2.3 These error estimates are not sharp, and the numerical results in Section 5 show a much better order of convergence.

3 The discretization space

We investigate here some properties of the spaceLν(D), which will be useful to study the mixed finite volume scheme. Recall thatL_ν(D) is the space of (u,v, F) which satisfy (7).

Lemma 3.1 [Poincar´e’s Inequality] Let us assume Assumption (2). Let D be an admissible discretization of Ω in the sense of Definition 2.1, such that regul(D) ≤θ for some θ >0. Let (ν_K)_K∈M be a family of nonnegative real numbers. Then there exists C₂ only depending on d, Ω and θsuch that, for all (u,v, F) ∈L_ν(D),

kukL²(Ω)≤C₂

kvkL²(Ω)^d+N₂(D, ν, F)

, (13)

where we have noted N₂(D, ν, F) =P

K∈M

P

σ∈EKdiam(K)^2d−2ν_K² F_K,σ² m(K)1/2

.

Proof.

Let R > 0 and x₀ ∈ Ω be such that Ω ⊂ B(x₀, R) (the open ball of center x₀ and radius R).

We extenduby the value 0 in B(x₀, R)\Ω, and we considerw∈H₀¹(B(x₀, R))∩H²(B(x₀, R)) such that−∆w(x) =u(x) for a.e. x∈B(x0, R). We multiply each equation of (7) byR

σ∇w(x)· n_K,σdγ(x), and we sum on σ∈ E; sincen_K,σ =−n_L,σ wheneverσ=K|L, we find

X

σ∈Eint,σ=K|L

vK·(xσ −xK) Z

σ∇w(x)·nK,σdγ(x) +vL·(xσ−xL) Z

σ∇w(x)·nL,σdγ(x)

+ X

σ∈Eext,σ∈EK

v_K·(x_σ−x_K) Z

σ∇w(x)·n_K,σdγ(x)

+ X

σ∈Eint,σ=K|L

ν_Km(K)F_K,σ Z

σ∇w(x)·n_K,σdγ(x) +ν_Lm(L)F_L,σ Z

σ∇w(x)·n_L,σdγ(x)

+ X

σ∈Eext,σ∈EK

ν_Km(K)F_K,σ Z

σ∇w(x)·n_K,σdγ(x)

= − X

σ∈Eint,σ=K|L

u_K Z

σ∇w(x)·n_K,σdγ(x) +u_L Z

σ∇w(x)·n_L,σdγ(x)

− X

σ∈Eext,σ∈EK

u_K Z

σ∇w(x)·n_K,σdγ(x).

Gathering by control volumes, we find X

K∈M

v_K· X

σ∈EK

(x_σ−x_K) Z

σ∇w(x)·n_K,σdγ(x)

+ X

K∈M

X

σ∈EK

ν_Km(K)F_K,σ Z

σ∇w(x)·n_K,σdγ(x) = − X

K∈M

u_K X

σ∈EK

Z

σ∇w(x)·n_K,σdγ(x)

= − X

K∈M

u_K Z

K

∆w(x)dx

= X

K∈M

m(K)u²_K =||u||²_L²_(Ω). (14) LetT₁ and T₂ be the two terms in the left-hand side of this equation.

Define T₃ =R

Ωv(x)· ∇w(x) dx; we have

|T₃| ≤ kvkL²(Ω)^dkwkH¹(Ω) (15)

and we want to compareT₁ withT₃. In order to do so, we apply Lemma 6.1 in the appendix to the vectorG_K = _m(K)¹ R

K∇w(x) dx, which gives Z

K∇w(x) dx= m(K)G_K = X

σ∈EK

m(σ)G_K·n_K,σ(x_σ−x_K) and therefore

T₃ = X

K∈M

v_K· X

σ∈EK

m(σ)G_K·n_K,σ(x_σ−x_K).

Hence, setting G_σ = _m(σ)¹ R

σ∇w(x) dγ(x), we get

|T₁−T₃| ≤ X

K∈M

|v_K| X

σ∈EK

m(σ) |G_K−G_σ|diam(K).

Thanks to the Cauchy-Schwarz inequality, we find (T₁−T₃)²≤



 X

K∈M

|v_K|² X

σ∈EK

m(σ)diam(K)







 X

K∈M

X

σ∈EK

m(σ)diam(K)|G_K−G_σ|²



.

We now apply Lemma 6.3 in the appendix, which givesC3 only depending ondandθsuch that

|G_K−G_σ|² ≤C₃diam(K)

m(σ) kwk²H²(K) (16)

(notice that α := ¹₂θ^−1/d < regul(D)^−1/d ≤ ρ_K/diam(K) is valid in Lemma 6.3). We also have, for σ ∈ EK, m(σ) ≤ω_d−1diam(K)^d−1, whereω_d−1 is the volume of the unit ball inR^d−1. Therefore, according to (6) and since regul(D)≥card(EK) for allK ∈ M,

(T₁−T₃)² ≤



 X

K∈M

|v_K|² X

σ∈EK

m(σ) diam(K)







 X

K∈M

X

σ∈EK

C₃diam(K)²||w||²_H²_(K)





≤ ω_d−1regul(D) X

K∈M

|v_K|²diam(K)^d

!

C₃size(D)²regul(D)kwk²_H²_(Ω)

≤ ω_d−1regul(D)²

ω_d ||v||²_L²_(Ω)^dC₃diam(Ω)²regul(D)kwk²_H²_(Ω). (17) Turning to T₂, we have T₂ = P

K∈M

P

σ∈EK ν_Km(K)F_K,σm(σ)G_σ ·n_K,σ, which we compare withT₄=P

K∈M

P

σ∈EKν_Km(K)F_K,σm(σ)G_K·n_K,σ thanks to (16):

(T2−T4)²

≤



 X

K∈M

X

σ∈EK

diam(K)m(σ)ν_K²F_K,σ² m(K)²







 X

K∈M

X

σ∈EK

m(σ)

diam(K)|G_K−G_σ|²





≤



ω_d−1ω_d X

K∈M

X

σ∈EK

diam(K)^2dν_K² F_K,σ² m(K)



regul(D)C₃||w||²_H²_(Ω)

≤ ω_d−1ω_ddiam(Ω)²N₂(D, ν, F)²regul(D)C₃||w||²H²(Ω). (18)

On the other hand, we can write T₄² ≤



 X

K∈M

X

σ∈EK

m(σ)²ν_K²F_K,σ² m(K)







 X

K∈M

X

σ∈EK

m(K)|G_K|²





≤ ω_d−1² N₂(D, ν, F)² regul(D) X

K∈M

m(K)|G_K|²

!

≤ ω_d−1² N₂(D, ν, F)²regul(D)kwk²H¹(Ω). (19) Thanks to (15), (17), (18) and (19), we can come back in (14) to find

||u||²_L²_(Ω) = T₁+T₂

≤ |T₁−T₃|+|T₃|+|T₂−T₄|+|T₄|

≤ s

ω_d−1C₃θ³

ω_d diam(Ω)kvkL²(Ω)^d||w||H²(Ω)+||v||L²(Ω)^d||w||H¹(Ω)

+p

ω_d−1ω_dC₃θdiam(Ω)N₂(D, ν, F)kwkH²(Ω)+ω_d−1√

θ N₂(D, ν, F)kwkH¹(Ω). Since there existsC4 only depending ondandB(x0, R) (the ball chosen at the beginning of the proof) such that ||w||H²(Ω) ≤C₄||u||L²(Ω), this concludes the proof.

Lemma 3.2 [Equicontinuity of the translations] Let us assume Assumption (2). Let D be an admissible discretization of Ω in the sense of Definition 2.1, such that regul(D) ≤θ for some θ >0. Let (ν_K)_K∈M be a family of nonnegative real numbers. Then there exists C₅ only depending on d, Ωand θ such that, for all (u,v, F) ∈L_ν(D) and all ξ∈R^d,

ku(·+ξ)−ukL¹(R^d)≤C₅

kvkL¹(Ω)^d+N₁(D, ν, F)

|ξ|, (20) where N₁(D, ν, F) = P

K∈M

P

σ∈EKdiam(K)^d−1ν_K|F_K,σ|m(K) (and u has been extended by 0 outside Ω).

Proof.

For all σ∈ E, let us define D_σu=|u_L−u_K|if σ =K|L and D_σu=|u_K|ifσ ∈ EK ∩ Eext. For (x, ξ) ∈ R^d×R^d and σ ∈ E, we define χ(x, ξ, σ) by 1 if σ∩[x, x+ξ] 6=∅ and by 0 otherwise.

We then have, for all ξ ∈ R^d and a.e. x ∈R^d (the x’s such that x and x+ξ do not belong to

∪K∈M∂K, and [x, x+ξ] does not intersect the relative boundary of any edge),

|u(x+ξ)−u(x)| ≤X

σ∈E

χ(x, ξ, σ)D_σu.

Applying (7), we get|u(x+ξ)−u(x)| ≤T₅(x) +T₆(x) with

T₅(x) = X

σ∈Eint,σ=K|L

χ(x, ξ, σ)(|v_K||x_σ−x_K|+|v_L||x_L−x_σ|)

+ X

σ∈Eext,σ∈EK

χ(x, ξ, σ)|vK||xσ−xK|

≤ X

K∈M

X

σ∈EK

χ(x, ξ, σ)diam(K)|vK|

and

T₆(x) = X

σ∈Eint,σ=K|L

χ(x, ξ, σ) (ν_Km(K)|F_K,σ|+ν_Lm(L)|F_L,σ|)

+ X

σ∈Eext,σ∈EK

χ(x, ξ, σ)ν_Km(K)|F_K,σ|

= X

K∈M

X

σ∈EK

χ(x, ξ, σ)ν_Km(K)|F_K,σ|.

In order that χ(x, ξ, σ) 6= 0, x must lie in the set σ−[0,1]ξ which has measure m(σ)|n_σ ·ξ| (where n_σ is a unit normal toσ). Hence,

Z

R^d

χ(x, ξ, σ)dx≤m(σ)|nσ·ξ| ≤ω_d−1diam(K)^d−1|ξ| ifσ ∈ EK. Since Card(EK)≤regul(D), this gives

Z

R^d

T₅(x)dx≤ω_d−1regul(D)|ξ| X

K∈M

diam(K)^d|v_K|

and Z

R^d

T₆(x) dx≤ω_d−1|ξ| X

K∈M

X

σ∈EK

diam(K)^d−1ν_K|F_K,σ|m(K), which concludes the proof thanks to (6).

Remark 3.1 We could prove the property

ku(·+ξ)−uk²_L²_(R^d₎≤C(kvk²_L²_(Ω)^d+N₂(D, ν, F)²) |ξ|(|ξ|+ size(D)),

by introducing the maximum value of diam(K)/ρ_L, for all (K, L) ∈ M × M, in the definition of regul(D). This would allow, in Theorem 2.2, to prove the strong convergence ofu_m in L²(Ω).

Nevertheless, we chose not to do so since the quantity diam(K)/ρL cannot remain bounded in a local mesh refinement procedure.

Note that we shall all the same prove, in a particular case, a strong convergence property in L²(Ω) for u, as a consequence of the error estimate.

Lemma 3.3 [Compactness property] Let us assume Assumption (2). Let (Dm)_m≥1 be admissible discretizations of Ω in the sense of Definition 2.1, such that size(Dm) → 0 as m→ ∞ and (regul(Dm))_m≥1 is bounded. Let(u_m,v_m, F_m, ν_m)_m≥1 be such that (u_m,v_m, F_m)∈ L_νm(Dm), (v_m)_m≥1 is bounded in L²(Ω)^d and N₂(Dm, ν_m, F_m) → 0 as m → ∞ (N₂ has been defined in Lemma 3.1).

Then there exists a subsequence of (Dm)_m≥1 (still denoted by (Dm)_m≥1) and u¯ ∈ H₀¹(Ω) such that the corresponding sequence (u_m)_m≥1 converges to u¯ weakly in L²(Ω)and strongly in L^q(Ω) for allq <2, and such that (v_m)_m≥1 converges to ∇u¯ weakly inL²(Ω)^d.

Proof.

Notice first that, for all discretizationD, for allν = (ν_K)_K∈Mnonnegative numbers and for all F = (F_K,σ)_{K∈M, σ∈EK},

N₁(D, ν, F) = X

K∈M

X

σ∈EK

diam(K)^d−1ν_K|F_K,σ|m(K)

≤



 X

K∈M

X

σ∈EK

diam(K)^2d−2ν_K²F_K,σ² m(K)





1/2

 X

K∈M

X

σ∈EK

m(K)





1/2

≤ N₂(D, ν, F)regul(D)^1/2m(Ω)^1/2. (21) Hence, ifN₂(D, ν, F) and regul(D) are bounded, so isN₁(D, ν, F). Owing to this, the hypotheses, Lemmas 3.1, 3.2 and Kolmogorov’s compactness theorem allow to extract a subsequence such that v_m → v¯ weakly in L²(Ω)^d and u_m → u¯ weakly in L²(Ω) and strongly in L¹(Ω) (which implies the strong convergence in L^q(Ω) for all q < 2). We now extend u_m, ¯u,v_m and ¯v by 0 outside Ω and we prove that ¯v=∇u¯ in the distributional sense on R^d. This will conclude that

¯

u∈H¹(R^d) and, since ¯u= 0 outside Ω, that ¯u∈H₀¹(Ω).

Let e ∈ R^d and ϕ ∈ C_c^∞(R^d). For simplicity, we drop the index m for Dm, v_m and u_m. We multiply each equation of (7) byR

σϕ(x) dγ(x)e·n_K,σ, we sum all these equations and we gather by control volumes, gettingT₇+T₈=T₉ with

T₇ = X

K∈M

v_K· X

σ∈EK

Z

σ

ϕ(x) dγ(x)e·n_K,σ(x_σ−x_K),

T8= X

K∈M

X

σ∈EK

νKm(K)FK,σ

Z

σ

ϕ(x) dγ(x)e·nK,σ

and

T₉ =− X

K∈M

u_K X

σ∈EK

Z

σ

ϕ(x) dγ(x) e·n_K,σ=− Z

Ω

u(x)div(ϕ(x)e) dx.

We want to compare T7 withT10 defined by T₁₀= X

K∈M

v_K · X

σ∈EK

1 m(K)

Z

K

ϕ(x) dxm(σ) e·n_K,σ(x_σ−x_K).

Since there exists C₆ only depending onϕsuch that, for all σ∈ EK,

1 m(σ)

Z

σ

ϕ(x) dγ(x)− 1 m(K)

Z

K

ϕ(x) dx

≤C₆size(D), we get that

|T₇−T₁₀| ≤C₆|e|size(D) X

K∈M

|v_K| X

σ∈EK

m(σ)|x_σ−x_K|.

But m(σ)|x_σ −x_K| ≤ ω_d−1diam(K)^d ≤ ^{ωd−1regul(D)}_ωd m(K) and, since card(EK) ≤regul(D), we obtain

|T₇−T₁₀| ≤C₆|e|size(D)ω_d−1regul(D)²

ω_d kvkL¹(Ω)

and thus lim_size(D)→0|T₇−T₁₀|= 0. Moreover, thanks to Lemma 6.1, we getT₁₀=R

Ωϕ(x)v(x)· edxand so lim_size(D)→0T₁₀=R

Ωϕ(x)¯v(x)·edx=R

R^dϕ(x)¯v(x)·edx (¯v has been extended by 0 outside Ω). This proves that

sizelim(D)→0T₇= Z

R^d

ϕ(x)¯v(x)·edx. (22)

Since ϕis bounded, by (21) we findC₇ only depending onϕand esuch that

|T8| ≤ C7

X

K∈M

X

σ∈EK

m(σ)νK|FK,σ|m(K)

≤ C₇ω_d−1N₁(D, ν, F)

≤ C₇ω_d−1regul(D)^1/2m(Ω)^1/2N₂(D, ν, F) and therefore, by the assumptions,

sizelim(D)→0T₈= 0. (23)

We clearly have

sizelim(D)→0T₉=− Z

Ω

¯

u(x)div(ϕ(x)e) dx =− Z

R^d

¯

u(x)div(ϕ(x)e) dx

(recall that ¯u has been extended by 0 outside Ω). Gathering this limit with (22) and (23) in T₇+T₈ =T₉, we obtain

Z

R^d

ϕ(x)¯v(x)·edx=− Z

R^d

¯

u(x)div(ϕ(x)e) dx,

which concludes the proof that ¯v=∇u¯ in the distributional sense on R^d.

4 Study of the mixed finite volume scheme

We first prove an a priori estimate on the solution to the scheme.

Lemma 4.1 Let us assume Assumptions (2)-(4). LetD be an admissible discretization of Ωin the sense of Definition 2.1. Let (ν_K)_K∈M be a family of positive real numbers and (u,v, F) ∈ L_ν(D) be a solution of ((7),(8),(9),(10)). Then, for allν₀ >0, for allβ₀≥β≥2−2dsuch that ν_K ≤ν₀diam(K)^β (∀K∈ M) and for all θ≥regul(D), this solution satisfies

kvk²_L²_(Ω)^d+ X

K∈M

X

σ∈EK

ν_KF_K,σ² m(K)≤C₈||f||²L²(Ω) (24) where C₈ only depends on d, Ω, α₀, θ, ν₀ and β₀.

Remark 4.1 This estimate shows that iff = 0thenF = 0andv= 0, and thusu= 0by Lemma 3.1. Since ((7),(8),(9),(10)) is square and linear in (u,v, F), the existence and uniqueness of the solution to the mixed finite volume scheme (i.e. Theorem 2.1) is an immediate consequence of this lemma.

Proof.

Multiply (10) byu_K, sum on the control volumes and gather by edges using (8):

X

σ∈Eint,σ=K|L

F_K,σ(u_L−u_K) + X

σ∈Eext,σ∈EK

−F_K,σu_K = Z

Ω

f(x)u(x)dx.

Using (7) and (8), and gathering by control volumes, this gives Z

Ω

f(x)u(x)dx

= X

σ∈Eint,σ=K|L

F_K,σv_K·(x_σ−x_K) +F_L,σv_L·(x_σ−x_L) + X

σ∈Eext,σ∈EK

F_K,σv_K·(x_σ−x_K)

+ X

σ∈Eint,σ=K|L

ν_Km(K)F_K,σ² +ν_Lm(L)F_L,σ² + X

σ∈Eext,σ∈EK

ν_Km(K)F_K,σ²

= X

K∈M

vK· X

σ∈EK

FK,σ(xσ−xK) + X

K∈M

X

σ∈EK

νKm(K)F_K,σ² . Applying (9), we obtain

Z

Ω

v(x)·Λ(x)v(x) dx+ X

K∈M

X

σ∈EK

ν_KF_K,σ² m(K) = Z

Ω

f(x)u(x) dx (25)

≤ ||f||L²(Ω)||u||L²(Ω). Using Young’s inequality and Lemma 3.1, we deduce that, for all ε >0,

α₀||v||²_L²_(Ω)^d+ X

K∈M

X

σ∈EK

ν_KF_K,σ² m(K)≤ 1

2ε||f||²L²(Ω)+εC₂²||v||²_L²_(Ω)^d +εC₂² X

K∈M

X

σ∈EK

diam(K)^2d−2ν_K² F_K,σ² m(K). (26) Since ν_K ≤ ν₀diam(K)^β, we have ν_Kdiam(K)^2d−2 ≤ν₀diam(K)^β+2d−2 ≤ ν₀diam(Ω)^β+2d−2 ≤ ν₀sup(1,diam(Ω)^β0+2d−2) (recall that β+ 2d−2≥0). Hence, (26) gives

α₀||v||²_L²_(Ω)^d+ X

K∈M

X

σ∈EK

ν_KF_K,σ² m(K)≤ 1

2ε||f||²L²(Ω)+εC₂²||v||²_L²_(Ω)^d +εν0sup(1,diam(Ω)^β0+2d−2)C₂² X

K∈M

X

σ∈EK

νKF_K,σ² m(K).

Taking ε= min(_2C^α02 2

,_2ν ¹

0sup(1,diam(Ω)^{β0 +2d−2})C₂²) concludes the proof of the lemma.

We now prove the convergence of the approximate solution toward the weak solution of (1).

Proof of Theorem 2.2.

For the simplicity of the notations, we omit the indexmas in the proof of Lemma 3.3. We first note that, thanks to Estimate (24) and sinceν_K =ν₀diam(K)^β,

N2(D, ν, F)² = X

K∈M

X

σ∈EK

diam(K)^2d−2ν_K²F_K,σ² m(K)

= ν₀ X

K∈M

X

σ∈EK

diam(K)^β+2d−2ν_KF_K,σ² m(K)

≤ ν₀size(D)^β+2d−2C₉

where C₉ does not depend on the discretization D (recall that regul(D) is bounded). Since β+ 2d−2>0, this last quantity tends to 0, and so doesN₂(D, ν, F), as size(D)→0. Hence, still using (24), we see that the assumptions of Lemma 3.3 are satisfied; there exists thus ¯u∈H₀¹(Ω) such that, up to a subsequence and as size(D) → 0, v → ∇u¯ weakly in L²(Ω)^d and u → u¯ weakly inL²(Ω) and strongly inL^q(Ω) for q <2.

We now prove that the limit function ¯u is the weak solution to (1). Since any subsequence of (u,v) has a subsequence which converges as above, and since the reasoning we are going to make proves that any such limit of a subsequence is the (unique) weak solution to (1), this will conclude the proof, except for the strong convergence ofv.

Let ϕ∈C_c^∞(Ω). We multiply (10) byϕ(x_K) and we sum on K. Gathering by edges thanks to (8), we get

X

σ∈Eint,σ=K|L

F_K,σ(ϕ(x_L)−ϕ(x_K)) = X

K∈M

Z

K

ϕ(x_K)f(x) dx

as long as size(D) is small enough (so thatϕ= 0 on the control volumesK such that∂K∩∂Ω6=

∅). We set, for σ=K|L, ϕ(x_L)−ϕ(x_K) = 1

m(K) Z

K∇ϕ(x) dx·(x_σ −x_K) + 1 m(L)

Z

L∇ϕ(x) dx·(x_L−x_σ) +R_KL and we have|R_KL| ≤C_ϕ(diam(K)²+ diam(L)²). We then obtain, gathering by control volumes and using (9) (and the fact that ϕ= 0 on the control volumes on the boundary of Ω),

Z

Ω

Λ_Dv(x)· ∇ϕ(x) dx= Z

Ω

f(x)ϕ_D(x) dx+T₁₁, (27) where Λ_D and ϕ_D are constant respectively equal to Λ_K and ϕ(x_K) on each meshK, and

|T₁₁| ≤C_ϕ X

σ∈Eint,σ=K|L

|F_K,σ|(diam(K)²+ diam(L)²) =C_ϕ X

K∈M

X

σ∈EK

diam(K)²|F_K,σ|. Let us estimate this term. We have

|T₁₁|² ≤ C_ϕ²



 X

K∈M

X

σ∈EK

ν_KF_K,σ² m(K)







 X

K∈M

X

σ∈EK

diam(K)⁴ ν_Km(K)





≤ C₁₀ X

K∈M

X

σ∈EK

diam(K)⁴

νKm(K)²m(K) (28)

where, according to (24), C₁₀ does not depend on the mesh since regul(D) stays bounded. But ν_K =ν₀diam(K)^β and diam(K)^d≤ ^regul(D)_ωd m(K), so that

diam(K)⁴

ν_Km(K)² ≤ regul(D)²diam(K)^4−β

ω²_dν₀diam(K)^2d = regul(D)²

ω²_dν₀ diam(K)^4−2d−β. Since 4−2d−β >0, we deduce from (28) that

|T11|²≤C10regul(D)²

ω²_dν₀ size(D)^4−2d−β X

K∈M

X

σ∈EK

m(K)≤ C10regul(D)³m(Ω)

ω²_dν₀ size(D)^4−2d−β

and this quantity tends to 0 as size(D)→0. Hence, we can pass to the limit in (27) to see that Z

Ω

Λ∇u(x)¯ · ∇ϕ(x) dx= Z

Ω

f(x)ϕ(x) dx , which proves that ¯u is the weak solution to (1).

The strong convergence of v to ∇u¯ is a consequence of (25). From this equation, and defining N(w)² =R

ΩΛ(x)w(x)·w(x) dx, we have N(v)²≤R

Ωf(x)u(x) dx and thus lim sup

size(D)→0

N(v)²≤ lim size(D)→0

Z

Ω

f(x)u(x) dx= Z

Ω

f(x)¯u(x) dx=N(∇u)¯ ² (29) (we use the fact that u → u¯ weakly in L²(Ω) and that ¯u is the weak solution to (1)). But N is a norm on L²(Ω)^d, equivalent to the usual norm and coming from the scalar product hw,zi = R

Ω

Λ(x)+Λ(x)^T

2 w(x)·z(x)dx; since v → ∇u¯ weakly in L²(Ω)^d as size(D) → 0, we therefore also haveN(∇u)¯ ≤lim inf_size(D)→0N(v). We conclude with (29) thatN(v)→N(∇u)¯ as size(D)→0, and thus that the weak convergence ofv to ∇u¯inL²(Ω)^d is in fact strong.

Remark 4.2 As a consequence of (25) and the strong convergence of v to ∇u, we see that¯ P

K∈M

P

σ∈EKν_KF_K,σ² m(K) → 0 as size(D) → 0. This strengthens Lemma 4.1 which only states that this quantity is bounded.

To conclude this section, we prove the error estimates. Note that these estimates could be extended, ford≤3, to the case ¯u∈H²(Ω) following some arguments of [15].

Proof of Theorem 2.3.

In this proof, we denote by C_i (for all integer i) various real numbers which can depend ond, Ω, ¯u, Λ and θ, but not on size(D). We also denote, for all K ∈ M and σ ∈ EK, ¯u_K = ¯u(x_K),

¯

u_σ = ¯u(x_σ),

F¯_K,σ = Z

σ

Λ(x)∇u(x)¯ ·n_K,σdγ(x), v¯K= 1

m(K)Λ⁻¹_K X

σ∈EK

F¯K,σ(xσ−xK)

(notice that Λ_K is indeed invertible since, from (3), Λ_K ≥α₀). Thanks to Lemma 6.1, we have

|v¯_K− ∇u(x)¯ | ≤C₁₁diam(K), ∀x∈K , ∀K ∈ M, (30) which implies

¯

v_K·(x_σ−x_K) = ¯u_σ −u¯_K+R_K,σ, ∀K∈ M, ∀σ∈ EK,

with |R_K,σ| ≤C₁₂diam(K)² for allK ∈ M and σ ∈ EK. Since ¯u is a classical solution to (1), we have

− X

σ∈EK

F¯_K,σ = Z

K

f(x) dx, ∀K∈ M.

Denoting, for allK ∈ Mand allσ ∈ EK,ub_K =u_K−u¯_K,bv_K=v_K−v¯_KandFb_K,σ =F_K,σ−F¯_K,σ, we see that

− X

σ∈EK

FbK,σ= 0, ∀K ∈ M, (31)

Fb_K,σ+Fb_L,σ = 0, ∀σ=K|L∈ Eint, (32) m(K)Λ_Kvb_K = X

σ∈EK

Fb_K,σ(x_σ−x_K), ∀K∈ M, (33) b

v_K·(x_σ−x_K) +vb_L·(x_L−x_σ) +ν_Km(K)Fb_K,σ+ν_Km(K) ¯F_K,σ+R_K,σ

−ν_Lm(L)Fb_L,σ−ν_Lm(L) ¯F_L,σ−R_L,σ=ub_L−ub_K,

∀K ∈ M, ∀L∈ NK, withσ =K|L, b

v_K·(x_σ−x_K) +ν_Km(K)Fb_K,σ+ν_Km(K) ¯F_K,σ+R_K,σ=−bu_K,

∀K ∈ M, ∀σ∈ EK∩ Eext.

(34)

We then get, multiplying (31) by bu_K, (34) byFb_K,σ and using (32), (33), X

K∈M

m(K)Λ_Kbv_K·bv_K+ X

K∈M

X

σ∈EK

ν_Km(K)Fb_K,σ² =T₁₂+T₁₃, (35) where

T₁₂=− X

K∈M

X

σ∈EK

ν_Km(K) ¯F_K,σFb_K,σ, T₁₃=− X

K∈M

X

σ∈EK

R_K,σFb_K,σ.

Using Young’s inequality and the fact that|F¯_K,σ| ≤C₁₃diam(K)^d−1, we have

|T₁₂| ≤ 1 2

X

K∈M

X

σ∈EK

ν_Km(K) ¯F_K,σ² +1 2

X

K∈M

X

σ∈EK

ν_Km(K)Fb_K,σ²

≤ C₁₄size(D)^β+2d−2+ 1 2

X

K∈M

X

σ∈EK

ν_Km(K)Fb_K,σ² .

Similarly, since |R_K,σ| ≤C₁₂diam(K)²,

|T₁₃| ≤ 1 2

X

K∈M

X

σ∈EK

R²_K,σ νKm(K) +1

2 X

K∈M

X

σ∈EK

ν_Km(K)Fb_K,σ²

≤ C₁₂² X

K∈M

X

σ∈EK

diam(K)⁴

ν_Km(K)²m(K) +1 2

X

K∈M

X

σ∈EK

νKm(K)Fb_K,σ²

≤ C₁₅size(D)^4−2d−β +1 2

X

K∈M

X

σ∈EK

ν_Km(K)Fb_K,σ² .

Gathering these two estimates in (35), the terms involving Fb_K,σ in the left-hand side and the right-hand side compensate and we obtain

α₀||bv||²_L²_(Ω)^d ≤C₁₆

size(D)^β+2d−2+ size(D)^4−2d−β

. (36)

Estimate (11) follows, using the fact that size(D)≤1 and that||v¯− ∇u¯||L^∞(Ω)^d ≤C₁₁size(D).