Finite Volume MethodsSchemes and Analysis

course at the University of Wroclaw

Robert Eymard1, Thierry Gallou et2 and Raphaele Herbin3

April, 2008

1Universite Paris-Est Marne-la-Vallee2Universite de Provence3Universite de Provence

Contents

Introduction 3

1 Finite volume discretizationof balance laws 41.1 A generic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.2 Finite volume meshes and space-time discretizations . . . . . . . . . . . . . . . . . . . . . . . .5

2 Linear diffusion problems 92.1 Actual problems and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192.3 Further analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

2.3.1 The convergence proof in the (harmonic averaging) isotropic case . . . . . . . . . . . . .252.3.2 The convergence proof in the case of the gradient schemes . . . . . . . . . . . . . . . . .272.3.3 The convergence proof in the case of the mixed-type schemes . . . . . . . . . . . . . . .33

3 Nonlinear convection - degenerate diffusion problems 343.1 Actual problems and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34

3.1.1 Finite volume approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .353.1.2 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

3.2 Further analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373.2.1 Continuous definitions and main convergence result . . . . . . . . . . . . . . . . . . . . .373.2.2 Existence, uniqueness and discrete properties . . . . . . . . . . . . . . . . . . . . . . . .383.2.3 Compactness of a family of approximate solutions . . . . . . . . . . . . . . . . . . . . .453.2.4 Convergence towards an entropy process solution . . . . . . . . . . . . . . . . . . . . . .463.2.5 Uniqueness of the entropy process solution. . . . . . . . . . . . . . . . . . . . . . . . . .503.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57

4 Navier-Stokes equations 584.1 Actual problems and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .584.2 Finite volume scheme with staggered variables . . . . . . . . . . . . . . . . . . . . . . . . . . .59

4.2.1 The MAC scheme on regular grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604.2.2 An example, built on the pressure grid . . . . . . . . . . . . . . . . . . . . . . . . . . . .614.2.3 Further analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62

4.3 Finite volume scheme with collocated variables . . . . . . . . . . . . . . . . . . . . . . . . . . .654.3.1 Discrete scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .654.3.2 Further analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .674.3.3 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

1

2

5 Discrete functional analysis 705.1 The topological degree argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .705.2 Discrete Sobolev embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

5.2.1 Discrete embedding ofW 1,1(Ω) in L1?

(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . 705.2.2 Discrete embedding ofW 1,p(Ω) in Lp?

(Ω), 1 < p < d . . . . . . . . . . . . . . . . . . . 725.2.3 Discrete embedding ofW 1,p(Ω) in Lq(Ω), for someq > p . . . . . . . . . . . . . . . . . 73

5.3 Compactness results for bounded families in discreteW 1,p(Ω) norm . . . . . . . . . . . . . . . . 735.3.1 Compactness inLp(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.2 Regularity of the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74

5.4 Properties in the case of the∆−adapted discretizations . . . . . . . . . . . . . . . . . . . . . . .765.5 Lemma used for time translates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81

Bibliography 83

Introduction

Our aim is to describe and analyse numerical schemes, issued from cell centred finite volume methods, appliedto various physical problems. For each problem, we present a mathematical model and some relevant finite vol-ume schemes. The efficiency of the schemes is then illustrated by some numerical results. In a second step, amathematical analysis of the schemes is carried out.

The first chapter introduces the notion of mesh, suitable for finite volume discretizations. In the second and thirdchapters, we focus on problems which are modeled by a scalar partial differential equation: we shall successivelydeal with pure diffusion operators, isotropic or not, and with convection-diffusion operators including the purehyperbolic 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 ofthe schemes.

3

Chapter 1

Finite volume discretizationof 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 discretiza-tions. LetΩ be a polygonal open subset ofRd, T ∈ R, and let us consider a scalar balance law written under thegeneral form:

ut + div(F (u,∇u)) + s(u) = 0 onΩ× (0, T ), (1.1)

whereF ∈ C1(R×Rd,Rd) ands ∈ C(R,R), andu is a real function of(x, t) ∈ Ω× (0, T ). In order to give themain ideas of the finite volume schemes, we need to introduce the setM containing the control volumes, which arethe subsets ofΩ on which balance equations are written (see Definition 1.1). Such a balance equation is obtainedfrom the above conservation law by integrating it over a control volumeK ∈M and applying the Stokes formula:

∫

K

ut dx +∫

∂K

F (u,∇u) · nK dγ(x) +∫

K

s(u) dx = 0,

wherenK stands for the unit normal vector to the boundary∂K outward toK andγ denotes the integration withrespect to the(d − 1)–dimensional Lebesgue measure. Let us denote byE the set of edges (faces in 3D) of themesh, andEK the set of edges which form the boundary∂K of the control volumeK. With these notations, theabove equation reads:

∫

K

ut dx +∑

σ∈EK

∫

σ

F (u,∇u) · nK dγ(x) +∫

K

s(u) dx = 0.

Let δt = T/M , whereM ∈ 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 onthe type of equation). We then get:

∫

K

u(n+1) − u(n)

δtdx +

∑

σ∈EK

∫

σ

F (u(n),∇u(n)) · nK dγ(x) +∫

K

s(u(n)) dx = 0,

whereu(n) denotes an approximation ofu(·, t(n)), with t(n) = nδt (see Definition 1.6 for a general definitionof a time discretization). Let us then introduce the discrete unknowns (one per control volume and time step)(u(n)

K )K∈M, n∈N; assuming the existence of such a set of real values, we may define a piecewise constant functionby:

u(n)M ∈ HM(Ω) : u

(n)M =

∑

K∈Mu

(n)K 1K ,

4

5

whereHM(Ω) denotes the space of functions fromΩ toR which are constant on each control volume of the meshM. In order to define the scheme, the fluxes

∫σ

F (u(n),∇u(n)) ·nK dγ(x) need to be approximated as a function

of the discrete unknowns. We denote byFK,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 fluxF (u,∇u) reducesto F (u,∇u) = vu, v ∈ Rd, and s(u) = 0:

ut + div(vu) = 0 onΩ. (1.2)

In order to approximate the fluxvu · n on the edges of the mesh, one needs to approximate the value ofu onthese edges, as a function of the discrete unknownsuK associated to each control volumeK. This may be donein several ways. A straightforward choice is to approximate the value ofu on the edgeσ = K|L separating thecontrol volumesK andL by the mean value12 (uK + uL). This yields the following numerical flux:

F(cv,c)K,σ (uM) = vK,σ

uK + uL

2

wherevK,σ =∫

σv · nK,σ, andnK,σ denotes the unit normal vector to the edgeσ outward toK. 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)K,σ (uM) = v+

K,σuK − v−K,σuL, (1.3)

wherex+ = max(x, 0) andx− = −min(x, 0). Hence we see that for purely convective problems, we do notrequire many geometrical properties on the mesh. Such meshes are formally defined in definition 1.1, where weonly 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) withF (u,∇u) =−Λ∇u + vu, v ∈ Rd, ands(u) = bu, b ∈ R:

ut + div(−Λ∇u + vu) + bu = 0 onΩ, (1.4)

the flux through a given edge then reads:∫

σ

F (u) · nK,σ =∫

σ

(−Λ∇u + v u) · nK,σ,

so that we now need to discretize the additional term∫

σΛ∇u ·nK,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 thatu is evaluated at some particular points of the mesh. This is the purposeof Definition of ”pointed meshes” (see Definition 1.2 and the following ones, each of them suited with particularmeshes 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 dis-cretizations, used all along this book for defining the different schemes. General finite volume meshes are used forpartial differential equations including first order, second order (scarcely more). In the case of convection prob-lems, we only need of a partition of the domain in control volumes, whereas in the case of diffusion problems, wealso 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?,Ω ⊂ Rd is open, polygonal, bounded, connected,∂Ω = Ω \ Ω is the boundary ofΩ, assumed to be Lipschitz-continuous.

(1.5)

6

In the above definition, polygonal means that∂Ω is a finite union of subsets of hyperplanes ofRd.Let us give a definition for a polygonal finite volume discretisation ofΩ.

Definition 1.1 (Polygonal finite volume space discretization ofΩ)Under hypothesis (1.5)a polygonal finite volume discretization ofΩ, is given by the pair(M, E), where:

1. M is a finite family of non empty connected open disjoint subsets ofΩ (the “control volumes”) such thatΩ = ∪K∈MK. For anyK ∈ M, let ∂K = K \K be the boundary ofK, m(K) > 0 denote the measureof K andhK denote the diameter ofK.

2. E is a finite family of disjoint subsets ofΩ (the “edges” of the mesh), such that, for allσ ∈ E , σ is a nonempty open subset of a hyperplane ofRd, whose (d-1)-dimensional measurem(σ) is stricly positive. Weassume that, for allK ∈ M, there exists a subsetEK of E such that∂K = ∪σ∈EK

σ. We then denote byMσ = K ∈ M, σ ∈ EK. We then assume that, for allσ ∈ E , eitherMσ has exactly one element andthenσ ⊂ ∂Ω (the set of these edges, called boundary edges, is denoted byEext) or Mσ has exactly twoelements (the set of these edges, called interior edges, is denoted byEint). For all K ∈ M andσ ∈ EK , wedenote for a.e.x ∈ σ bynK,σ the unit vector normal toσ outward toK.

For all K ∈M, we denote byNK the set of the neighbours ofK:

NK = L ∈M \ K, ∃σ ∈ Eint, Mσ = K,L. (1.6)

We then denote byHM(Ω) the set of functions fromΩ to R, constant on each element ofM. The size of thediscretization is defined by:

h(M) = supdiam(K),K ∈M. (1.7)

We then define the spaceXD = RM×E , containing all the families of reals((uK)K∈M, (uσ)σ∈E), and weconsider the subspaceXD,0 of all families((uK)K∈M, (uσ)σ∈E) such thatuσ = 0 if σ ∈ Eext. We then define,for u ∈ XD, ΠMu ∈ HM(Ω) as the element ofHM(Ω) equal a.e. inK to the valueuK , for all K ∈M.

In order to study the schemes introduced in this paper for diffusion operators, we also define the notion of pointedfinite volume discretization.

Definition 1.2 (Pointed polygonal finite volume space discretization ofΩ)Under hypothesis (1.5), a pointed polygonal finite volume discretization ofΩ, is given by the triplet(M, E ,P),where:

1. (M, E) is a polygonal finite volume discretization ofΩ in the sense of definition 1.1.

2. P is a family of points ofΩ indexed byM andE , denoted byP = ((xK)K∈M, (xσ)σ∈E), such that for allK ∈ M, xK ∈ K and for all σ ∈ E , xσ ∈ σ. We then denote bydK,σ the abscissae ofxK on the straightline with originexσ and oriented bynK,σ.

We then definePM : C(Ω) → HM(Ω) by

PMϕ(x) = ϕ(xK) for a.e.x ∈ K, ∀K ∈M, ∀ϕ ∈ C(Ω). (1.8)

andPD : C(Ω) → XD by

PDϕK = ϕ(xK), ∀K ∈M, PDϕσ = ϕ(xσ), ∀σ ∈ E , ∀ϕ ∈ C(Ω). (1.9)

Remark 1.1 The conditionsxK ∈ K andxσ ∈ σ can easily be relaxed.

7

xK

dKσ

xσ

Figure 1.1: Pointed mesh

Thanks to the algebraic definition ofdK,σ, we have the relation

∑

σ∈EK

m(σ)dK,σ = d m(K), ∀K ∈M. (1.10)

Definition 1.3 (Pointed star-shaped polygonal finite volume space discretization ofΩ)Under hypothesis (1.5), a pointed polygonal finite volume discretization ofΩ, denoted by(M, E ,P), is said to bestar-shaped, if allK ∈ M is xK-star-shaped, which means that for allx ∈ K, the property[xK , x] ⊂ K holds(this is equivalent todK,σ ≥ 0 for all σ ∈ EK). A pointed polygonal finite volume discretization ofΩ, denoted by(M, E ,P), is said to be strictly star-shaped if, for allK ∈M andσ ∈ EK , dK,σ > 0.For all K ∈M andσ ∈ EK , we denote byDK,σ the cone with vertexxK and basisσ

DK,σ = txK + (1− t)y, t ∈ (0, 1), y ∈ σ, (1.11)

and we then define the setLD(Ω) of all functionsg which are constant in each coneDK,σ (we then denote bygK,σ

the corresponding value). Note thatHM(Ω) ⊂ LD(Ω). We denote, for allσ ∈ E , Dσ =⋃

K∈MσDK,σ (this set

is sometimes called the “diamond” associated to the edgeσ). We then definedσ, for all σ ∈ E , by

dσ = dL,σ + dK,σ, ∀σ ∈ Eint, Mσ = K, L,dσ = dK,σ, ∀σ ∈ Eext, Mσ = K. (1.12)

Remark 1.2 The above definition applies to a large variety of meshes. Note that no hypothesis is made on theconvexity of the control volumes, which enables that “hexahedra” with non planar faces can be used (in fact, suchsets 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 ofΩ)Under hypothesis (1.5), a strictly star-shaped pointed polygonal finite volume discretization ofΩ, denoted by(M, E ,P), is said to be a∆−adapted pointed polygonal finite volume space discretization ofΩ, if, for all σ ∈ Eint,denoting byK, L ∈ M the two control volumes such thatMσ = K, L, then the straight line(xK ,xL) isorthogonal toσ. In this case, we denote byxσ the intersection betweenσ and the line(xK , xL) (assumed tobelong toσ for the sake of simplicity). Then, forK, L ∈ M, such that there existsσ ∈ Eint withMσ = K, L,σ is unique and one denotesσ = K|L. For all K ∈ M andσ ∈ EK ∩ Eext, we denote byxσ the orthogonalprojection ofxK onσ.

An example of two neighbouring control volumesK andL of M is depicted in Figure 1.2.

8

xL

xK

LK

m(σ)

K|L

dKL

dK,σ

Figure 1.2: Notations for a∆-adapted mesh

Definition 1.5 (∆−super-adapted pointed polygonal finite volume space discretization ofΩ) Under hypothe-sis (1.5), a ∆−adapted pointed polygonal finite volume discretization ofΩ, denoted by(M, E ,P), is said to be∆−super adapted, if, for allσ ∈ Eint, denoting byK,L ∈ M the two control volumes such thatMσ = K, L,then the straight line(xK , xL) is orthogonal toσ and intersectsσ at the pointxσ, equal to the center of gravityof σ.

Definition 1.6 (Time discretization of(0, T )) A time discretization of(0, T ) is given by an integer valueN andby an increasing sequence of real values(tn)n∈[[0,N+1]] with t0 = 0 and tN+1 = T . The time steps are thendefined byδtn = tn+1 − tn, for n ∈ [[0, N ]].

Definition 1.7 (Space-time discretization ofΩ× (0, T )) A finite volume discretizationD of Ω × (0, T ) is thefamily D = (M, E , (xK)K∈M, N, (tn)n∈[[0,N ]]), whereM, E , (xK)K∈M is a finite volume mesh ofΩ in thesense of one of the Definitions 1.1-1.5 andN , (tn)n∈[[0,N+1]] is a time discretization of(0, T ) in the sense ofDefinition 1.6. For a given meshD, one defines:

|D| = max(h(M), (δtn)n∈[[0,N ]])

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 solutionu of the following diffusionproblem with full anisotropic tensor:

−div(Λ∇u) = f in Ω,u = 0 on∂Ω,

(2.1)

under the following assumptions:

Ω is an open bounded connected polygonal subset ofRd, d ∈ N?, (2.2)

Λ is a measurable function fromΩ toMd(R), whereMd(R) denotes the set ofd× d matrices, such that for a.e.x ∈ Ω,Λ(x) is symmetric,the lowest and the largest eigenvalues ofΛ(x), denoted byλ(x) andλ(x),are such thatλ, λ ∈ L∞(Ω) and there existsλ0 ∈ R with0 < λ0 ≤ λ(x) ≤ λ(x) for a.e.x ∈ Ω,

(2.3)

andf ∈ L2(Ω). (2.4)

We give the classical weak formulation in the following definition.

Definition 2.1 (Weak solution) Under hypotheses (2.2)-(2.4), we say thatu is a weak solution of (2.1) if

u ∈ H10 (Ω),∫

Ω

Λ(x)∇u(x) · ∇v(x)dx =∫

Ω

f(x)v(x)dx, ∀v ∈ H10 (Ω). (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 thediscretization methods of all these terms are independent of one another, and the treatment of convection andreaction 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 unknownlocated at the edges of the mesh. A variety of situations with respect to a possible elimination of this unknown arethen 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

10

2. We wish to be able to prove the convergence of the discrete solution and an associate gradient to the solutionof the continuous problem and its gradient, and to show error estimates.

Under hypothesis (1.5), letD = (M, E ,P) be a discretization ofΩ in the sense of definition 1.2 (we shall alsoconsider 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 settingup a system of discrete equations for a family of values((uK)K∈M, (uσ)σ∈E) in the control volumes and on theinterfaces. 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 volumeK ∈M,which formally gives (assuming sufficient regularity onu andΛ) the following balance equation on the controlvolumeK: ∑

σ∈EK

(−

∫

σ

Λ(x)∇u(x) · nK,σdγ(x))

=∫

K

f(x)dx.

The flux− ∫σ

Λ(x)∇u(x)·nK,σdγ(x) is approximated by a functionFK,σ(u) of the values((uK)K∈M, (uσ)σ∈E)at the “centers” and at the interfaces of the control volumes (in all the cases presented below,FK,σ(u) only dependson uK and all(uσ′)σ′∈EK

). We give the definition of the setsXD andXD,0 (also provided in Definition (1.1))where the discrete unknowns lie, that is to say:

XD = v = ((vK)K∈M, (vσ)σ∈E), vK ∈ R, vσ ∈ R, (2.6)

XD,0 = v ∈ XD such thatvσ = 0 ∀σ ∈ Eext. (2.7)

A discrete equation corresponding to (2.1) is then: findu ∈ XD,0 such that

∑

σ∈EK

FK,σ(u) =∫

K

f(x)dx, ∀K ∈M. (2.8)

The valuesuσ on the interfaces are then introduced so as to allow for a consistent approximation of the normalfluxes in the case of an anisotropic operator and a general, possibly nonconforming mesh. Thanks to the defi-nition of XD,0, the valuesuσ for the boundary faces or edges are prescribed by the discrete counterpart of thehomogeneous Dirichlet boundary condition:

uσ = 0, ∀σ ∈ Eext. (2.9)

We then need card(Eint) equations to ensure that the problem is well posed. Following the finite volume ideas, wemay write the continuity of the discrete flux for all interior edges, that is to say:

FK,σ(u) + FL,σ(u) = 0, for σ ∈ Eint such thatMσ = K, L. (2.10)

We now have card(M) + card(Eint) unknowns and equations. Note thatu ∈ XD,0 is solution of (2.8)-(2.9)-(2.10)if and only if it satisfies

u ∈ XD,0,

〈u, v〉F =∑

K∈MvK

∫

K

f(x)dx, for all v ∈ XD,0,(2.11)

where we set〈u, v〉F =

∑

K∈M

∑

σ∈EK

FK,σ(u)(vK − vσ). (2.12)

Indeed, takingvK = 1, vL = 0 for all L ∈M with L 6= K andvσ = 0 for all σ ∈ E leads to (2.8). Takingvσ = 1for σ ∈ Eint, vσ′ = 0 for all σ′ ∈ E with σ′ 6= σ andvK = 0 for all K ∈ M provides (2.10). Reciprocally, for allv ∈ XD,0, multiplying (2.8) byvK 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, leadingto different properties for the resulting scheme. Let us first turn to the simple case of isotropic diffusion, using∆−adapted pointed meshes.

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 whereId is the identity application ofRd for a.e.x ∈ Ω. (2.13)

Let D be an admissible discretization ofΩ in the sense of Definition 1.4. The finite volume approximation toProblem (2.1) is given as the solutionu = ((uK)K∈M, (uσ)σ∈E) ∈ XD,0 of (2.8)-(2.9)-(2.10), in which we definethe numerical flux by:

FK,σ(u) = −m(σ)uσ − uK

dK,σ, ∀K ∈M, ∀σ ∈ EK . (2.14)

The mathematical analysis of this scheme is given in section 2.3.1. Nevertheless, let us rewrite the scheme, re-marking that we can immediately eliminateuσ for Mσ = K,L from the relation (2.10). Indeed we get

uσ =dL,σuK + dK,σuL

dL,σ + dK,σ. (2.15)

Hence we can identify the set of all elements ofXD,0, such that the condition (2.15) is ensured, with the setHM(Ω), defined in Definition 1.1 as the set of the piecewise constant functions in each control volume. Reportingthe above expression ofuσ given by (2.15) in (2.14), we get

FK,σ(u) = −m(σ)δK,σu

dσ, ∀K ∈M, ∀σ ∈ EK , (2.16)

where we define, for allu ∈ HM(Ω)

δK,σu = uL − uK , ∀σ ∈ Eint, Mσ = K,L,δK,σu = −uK , ∀σ ∈ Eext, Mσ = K, (2.17)

and where we recall thatdσ = dL,σ + dK,σ, ∀σ ∈ Eint, Mσ = K, L,dσ = dK,σ, ∀σ ∈ Eext, Mσ = K.

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 ∈ HM(Ω), such that

−∑

σ∈EK

m(σ)δK,σu

dσ=

∫

K

f(x)dx, ∀K ∈M.

Recall that the above scheme also writes

u ∈ HM(Ω),∑

K∈M

∑

σ∈EK

m(σ)dK,σδK,σu

dσ

δK,σv

dσ=

∫

Ω

f(x)v(x)dx, ∀v ∈ HM(Ω). (2.18)

The case of harmonic averaging of an isotropic tensor

We assume in this section the hypothesis (2.2) and that

Λ(x) = λ(x)Id whereId is the identity application ofRd

with λ ∈ L∞(Ω) such that there existsλ0 ∈ (0, +∞) withλ0 ≤ λ(x) for a.e.x ∈ Ω.

(2.19)

We consider in this section the so-called “harmonic averaging” case. LetD be an admissible∆−adapted dis-cretization ofΩ in the sense of Definition 1.4. The finite volume approximation to Problem (2.1) is given as thesolutionu = ((uK)K∈M, (uσ)σ∈E) ∈ XD,0 of (2.8)-(2.9)-(2.10), in which we define the numerical flux by:

FK,σ(u) = −m(σ)λKuσ − uK

dK,σ, ∀K ∈M, ∀σ ∈ EK . (2.20)

12

In (2.20), we set

λK =1

m(K)

∫

K

λ(x)dx, ∀K ∈M.

Note that, eliminatinguσ for all σ ∈ Eint leads to findu ∈ HM(Ω) such that

−∑

σ∈EK

λσm(σ)δK,σu

dσ=

∫

K

f(x)dx, ∀K ∈M, (2.21)

denoting byλσ, for all σ ∈ E , the harmonic averaged value defined by:

λσ = dσ

λK

dK,σ

λL

dL,σ

λK

dK,σ+ λL

dL,σ

, ∀σ ∈ Eint, Mσ = K, L

λσ = λK , ∀σ ∈ Eext, Mσ = K.(2.22)

Then the solution of (2.21) is also solution of

uD ∈ HM(Ω),∑

K∈M

∑

σ∈EK

λσm(σ)dK,σδK,σu

dσ

δK,σv

dσ=

∫

Ω

f(x)v(x)dx, ∀v ∈ HM(Ω). (2.23)

Elimination of uσ in the general case

In the case where the relation (2.19) does not hold, we are then led to look for an expression ofFK,σ(u) dependingonuK and all(uσ′)σ′∈EK . In this case, it is no longer possible to locally eliminateuσ for all σ ∈ Eint, 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(uσ)σ∈E as local affine combinations of the values(uK)K∈M [35] and therefore to eliminate them. The idea is to remark that the linear system constituted bythe equations (2.8) for all K ∈ MS , whereMS is the set of all simplices sharing the same interior vertexS,and (2.10) for all the interior edges such thatMσ ⊂ MS , presents as many equations as unknownsuσ, forσ ∈ ∪K∈MS

EK . Indeed, the number of edges in∪K∈MSEK such thatMσ 6⊂ MS is equal to the number of

control volumes inMS . Unfortunately, there is at this time no general result on the invertibility nor the symmetryof 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 forma(·, ·), but witha space of dimension smaller than that ofXD,0. This can be achieved by expressing the value ofu on any interiorinterfaceσ ∈ Eint as a consistent barycentric combination of the valuesuK :

uσ =∑

K∈MβK

σ uK , (2.24)

where(βKσ ) K∈M

σ∈Eint

is a family of real numbers, withβKσ 6= 0 only for some control volumesK close toσ, and such

that ∑

K∈MβK

σ = 1 andxσ =∑

K∈MβK

σ xK , ∀σ ∈ Eint. (2.25)

We recall that the valuesuσ, σ ∈ Eext are set to 0 in order to respect the boundary conditions. This ensures that ifϕ is a regular function, thenϕσ =

∑K∈M βK

σ ϕ(xK) is a consistent approximation ofϕ(xσ) for σ ∈ Eint. Hencethe new scheme reads:

Findu ∈ XD,0 such thatuσ =∑

K∈MβK

σ uK , ∀σ ∈ Eint, and

〈u, v〉F =∑

K∈MvK

∫

K

f(x)dx, for all v ∈ XD,0 with vσ =∑

K∈MβK

σ vK , ∀σ ∈ Eint.(2.26)

13

This method has been shown in [25] to be efficient in the case of a problem whereΛ = Id (for the approximation ofthe viscous terms in the Navier-Stokes problem). Unfortunately, because of a poor approximation of the local fluxat strongly hetereogenous interfaces, this approach is not sufficient to provide accurate results for some types offlows in heterogeneous media, as we shall show in section 2.2. This is especially true when using coarse meshes, asis often the case in industrial problems. Therefore it is possible to take advantage of both techniques: we shall useequation (2.12) and keep the unknownsuσ on the edges which require them, for instance those where the matrixΛ is discontinuous: hence (2.10) will hold for all edges associated to these unknowns; for all other interfaces, weshall impose the values ofu using (2.24), and therefore eliminate these unknowns. Let us decompose the setEint

of interfaces into two non intersecting subsets, that is:Eint = A∪ Ehyb, Ehyb = Eint \ A. The interface unknownsassociated withA 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 thematrixΛ is discontinuous are located within the set

⋃σ∈Ehyb

σ, such a property is not needed in the mathematicalstudy of the scheme.

Let us introduce the spaceXD,A ⊂ XD,0 defined by:

XD,A = v ∈ XD such thatvσ = 0 for all σ ∈ Eext andvσ satisfying(2.24) for all σ ∈ A. (2.27)

The composite scheme which we consider in this work reads:

Findu ∈ XD,A such that:〈u, v〉F =

∑K∈M vK

∫K

f(x)dx, for all v ∈ XD,A.(2.28)

We therefore obtain a scheme with card(M) + card(Ehyb) equations and unknowns. It is thus less expensive whileit remains precise (for the choice of numerical flux given below) even in the case of strong heterogeneities (seesection 2.2).Note that with the present scheme, (2.10) holds for allσ ∈ Ehyb, but not generally for anyσ ∈ A. However, fluxesbetween pairs of control volumes can nevertheless be identified. These pairs are no longer necessarily connectedby 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 discretespaceXD,A is modified to include the unknowns associated to the corresponding edges, and the resulting discreteweak 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

uσ =∑

K∈MβK

σ uK +∑

σ′∈Ehyb

βσ′σ uσ′ , ∀σ ∈ A,

with ∑

K∈MβK

σ +∑

σ′∈Ehyb

βσ′σ = 1 andxσ =

∑

K∈MβK

σ xK +∑

σ′∈Ehyb

βσ′σ xσ′ , ∀σ ∈ A.

Then all the mathematical properties shown below still hold.

Let us apply these principles in two situations. The first one is the case of an anisotropic tensor and of a∆−adaptedmesh, for which we shall be able to get back a standard finite volume scheme. The second one is that of ananisotropic tensor and of a general pointed mesh.

The case of an anisotropic tensor and of a∆−adapted mesh

Under hypotheses (2.2)-(2.4), letD be a∆−adapted finite volume discretization ofΩ in the sense of Definition1.4. For the sake of simplicity, we setA = Eint, consider the spaceXD,Eint ⊂ XD,0 defined by:

XD,Eint = v ∈ XD such thatvσ = 0 for all σ ∈ Eext andvσ =dK,σvL + dL,σvK

dK,σ + dL,σfor all σ ∈ Eint. (2.29)

14

Indeed, it is possible, in the particular case of a∆−adapted finite volume discretization ofΩ, to use the propertythat xσ is located, forMσ = K, L, on the line(xK , xL), and therefore, express (2.24) using only the twovaluesuK anduL.

We then introduce a discrete gradient∇D : XD,Eint → HM(Ω)d, given by constant values in each controlvolume:

∇Ku =1

m(K)

∑

σ∈EK

m(σ)(xσ − xK)uσ − uK

dK,σ, ∀K ∈M, ∀u ∈ XD,Eint . (2.30)

A first natural scheme, inspired by the finite element framework, would be

u ∈ XD,Eint ,∫

Ω

Λ(x)∇Du(x) · ∇Dv(x)dx =∫

Ω

f(x)ΠMv(x)dx, ∀v ∈ XD,Eint .

(recall thatΠMv is defined in Definition 1.1). But this scheme is not convenient, since it is possible to get∇Du = 0with u 6= 0. Hence a stabilization is necessary. To this purpose, we define an elementRDu ∈ LD(Ω), given bythe expression

RK,σu =√

αK d

(uσ − uK

dK,σ−∇Ku · nK,σ

), ∀K ∈M, ∀σ ∈ EK , ∀u ∈ HM(Ω). (2.31)

where(αK)K∈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(αK)K∈M,as the solution of the following equation:

u ∈ XD,Eint ,∫

Ω

(Λ(x)∇Du(x) · ∇Dv(x) + RDu(x)RDv(x)) dx =∫

Ω

f(x)ΠMv(x)dx, ∀v ∈ XD,Eint .(2.32)

Remark 2.6 In the caseΛ = Id, if D is a∆−superadapted discretization in the sense of definition 1.5 (recall thatit meansxσ−xK = dK,σnK,σ, for all K ∈M andσ ∈ EK) andαK = 1 for all K ∈M, then the scheme (2.32)gives back the scheme provided above for the Laplace case. Indeed, we get that

∑

σ∈EK

m(DK,σ)RK,σuRK,σv =∑

σ∈EK

m(σ)dK,σ

dd

(uσ − uK

dK,σ−∇Ku · nK,σ

)(vσ − vK

dK,σ−∇Kv · nK,σ

),

which leads to∑

σ∈EK

m(DK,σ)RK,σuRK,σv =∑

σ∈EK

m(σ)dK,σuσ − uK

dK,σ

vσ − vK

dK,σ−m(K)∇Ku · ∇Kv,

using∑

σ∈EKm(σ)dK,σ

uσ−uK

dK,σnK,σ = m(K)∇Ku and

∑σ∈EK

m(σ)dK,σnK,σntK,σ = m(K)Id.

In the sequel, we shall refer to the scheme (2.32) as the gradient scheme on adapted meshes, since it is obtainedfrom a discretization formula on the gradient, the consistence of which results from the orthogonality property ofthe mesh.

We may note that scheme (2.32) leads to∑

K∈M

(m(K)ΛK∇Ku · ∇Kv

+αK

∑

σ∈EK

m(σ)dK,σ

(uσ − uK

dK,σ−∇Ku · nK,σ

)(vσ − vK

dK,σ−∇Kv · nK,σ

))=

∑

K∈MvK

∫

K

f(x)dx,

setting

ΛK =1

m(K)

∫

K

Λ(x)dx.

15

This expression of the scheme can then be put under the form∑

K∈M

∑

σ∈EK

∑

σ′∈EK

Aσ′σK (uσ − uK)(vσ − vK) =

∑

K∈MvK

∫

K

f(x)dx,

where the matrix(Aσ′σK )σ′,σ∈EK

is symmetric. Let us define

FK,σ(u) = −∑

σ′∈EK

Aσ′σK (uσ′ − uK), ∀σ ∈ EK , ∀K ∈M. (2.33)

Taking forv ∈ XD,Eint , vK = 1 andvL = 0 for all L ∈M\K, we obtain that the above equation is equivalentto finding the values(uK)K∈M, solution of the following system of equations:

∑

σ∈EK

FK,σ(u) =∫

K

f(x)dx, ∀K ∈M, (2.34)

where

FK,σ(u) =dK,σ

dK,σ + dL,σFK,σ(u)− dL,σ

dK,σ + dL,σFL,σ(u), ∀σ ∈ Eint, Mσ = K, L (2.35)

andFK,σ(u) = FK,σ(u), ∀σ ∈ Eext, Mσ = K. (2.36)

This is indeed a finite volume scheme (in the sense of the respect of the local balance), since

FK,σ(u) = −FL,σ(u), ∀σ ∈ Eint, Mσ = K, 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. Thisis for instance the case in subsurface flow modelling, where parallelepipedic (or quadrangular in 2D) meshes arewidely used to mesh the underground layers. In the case of geologigal faults for instance, the independent meshingof geological layers results in non conforming meshes (see e.g. Figure 2.1) which are clearly not∆−adaptedmeshes.

Figure 2.1: Non conforming subsurface flow modelling

Hence we consider thatD = (M, E ,P) is only a discretization ofΩ in the sense of definition 1.2, andwe thendenote byxσ the center of gravity of σ ∈ E . As we did in the case of an anisotropic diffusion, on∆−adaptedmeshes, we again identify the numerical fluxesFK,σ(u) through the mesh dependent bilinear form〈·, ·〉F definedin (2.12), using the expression of a discrete gradient. Indeed let us assume that, for allu ∈ XD, we have constructeda discrete gradient∇Du, we then seek a family(FK,σ(u)) K∈M

σ∈EK

such that

∑

K∈M

∑

σ∈EK

FK,σ(u)(vK − vσ) =∫

Ω

∇Du(x) · Λ(x)∇Dv(x)dx, ∀u, v ∈ XD. (2.37)

16

Remark 2.7 It is then always possible to deduce an expression forFK,σ(u) satisfying (2.37), under the suffi-cient condition that, for allK ∈ M and a.e. x ∈ K, ∇Du(x) may be expressed as a linear combination of(uσ − uK)σ∈EK

, the coefficients of which are measurable bounded functions ofx. This property is ensured in theconstruction of∇Du(x) given below.

Then, in order to ensure the desired properties 2 and 3, we shall see in Section 2.3.2 that it suffices that the discretegradient satisfies the following properties.

1. For a sequence of space discretisations ofΩ with mesh size tending to 0, if the sequence of associated gridfunctions is bounded in some sense, then their discrete gradient is expected to converge, at least weakly inL2(Ω)d, to the gradient of an element ofH1

0 (Ω);

2. If ϕ is a regular function fromΩ toR, the discrete gradient of the piecewise function defined by taking thevalueϕ(xK) on each control volumeK andϕ(xσ) on each edgeσ is a consistent approximation of thegradient ofϕ.

Le us first define:

∇Ku =1

m(K)

∑

σ∈EK

m(σ)(uσ − uK)nK,σ, ∀K ∈M, ∀u ∈ XD, (2.38)

wherenK,σ is the outward toK normal unit vector,m(K) andm(σ) are the usual measures (volumes, areas, orlengths) ofK andσ. The consistency of formula (2.38) stems from the following geometrical relation:

∑

σ∈EK

m(σ)nK,σ(xσ − xK)t = m(K)Id, ∀K ∈M, (2.39)

where(xσ − xK)t is the transpose ofxσ − xK ∈ Rd, andId is thed × d identity matrix. Indeed, for any linearfunction defined onΩ by ψ(x) = G · x with G ∈ Rd, assuming thatuσ = ψ(xσ) anduK = ψ(xK), we getuσ − uK = (xσ − xK)tG = (xσ − xK)t∇ψ, hence (2.38) leads to∇Ku = ∇ψ.Since the coefficient ofuK in (2.38) is in fact equal to zero, a reconstruction of the discrete gradient∇Du solelybased on (2.38) cannot lead to a definite discrete bilinear form in the general case. Hence, we now introduce:

∇K,σu = ∇Ku + RK,σu nK,σ, (2.40)

with

RK,σu =

√d

dK,σ(uσ − uK −∇Ku · (xσ − xK)) , (2.41)

(recall thatd is the space dimension anddK,σ is the Euclidean distance betweenxK andσ). We may then define∇Du as the piecewise constant function equal to∇K,σu a.e. in the coneDK,σ with vertexxK and basisσ:

∇Du(x) = ∇K,σu for a.e.x ∈ DK,σ. (2.42)

We can then prove that the discrete gradient defined by (2.38)-(2.42) meets the required properties (see Lemmas2.6 and 2.7). In order to identify the numerical fluxesFK,σ(u) using the relation (2.37), we put the discrete gradientunder the form

∇K,σu =∑

σ′∈EK

(uσ′ − uK)yσσ′ ,

with

yσσ′ =

m(σ)m(K)

nK,σ +

√d

dK,σ

(1− m(σ)

m(K)nK,σ · (xσ − xK)

)nK,σ if σ = σ′

m(σ′)m(K)

nK,σ′ −√

d

dK,σm(K)m(σ′)nK,σ′ · (xσ − xK)nK,σ otherwise.

(2.43)

Thus:∫

Ω

∇Du(x) · Λ(x)∇Dv(x)dx =∑

K∈M

∑

σ∈EK

∑

σ′∈EK

Aσσ′K (uσ − uK)(vσ′ − vK), ∀u, v ∈ XD, (2.44)

17

with:

Aσσ′K =

∑

σ′′∈EK

yσ′′σ · ΛK,σ′′yσ′′σ′ andΛK,σ′′ =

∫

DK,σ′′Λ(x)dx. (2.45)

Then we get that the local matrices(Aσσ′K )σσ′∈EK are symmetric and positive, and the identification of the numer-

ical fluxes using (2.37) leads to the expression:

FK,σ(u) =∑

σ′∈EK

Aσσ′K (uK − uσ′). (2.46)

The properties provided by this definition (which could not have been obtained using natural expansions of regularfunctions) are shown in Lemma 2.8. Then Theorem 2.1 shows that these properties are sufficient to provide theconvergence of the scheme. Note that the proof of this property holds for general heterogeneous, anisotropic andpossibly discontinuous fieldsΛ, for which the solutionu of (2.5) is not in general more regular thanu ∈ H1

0 (Ω).The local consistency property provided by definition (2.46) is only detailed in the error estimate theorem 2.2, inthe case whereΛ andu are regular enough.

Remark 2.8 The choice of the coefficient√

d in (2.41) is not compulsory, and any fixed positive value could besubstituted; it is motivated by the fact that it provides a diagonal matrixAK in the case of isotropic diffisionand of meshes which satisfynK,σ = xσ−xK

dK,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 thatD = (M, E ,P) is only a discretization ofΩ in the sense of definition 1.2, andwe thendenote byxσ the center of gravity of σ ∈ E . Our purpose is again to define a relation between the numericalfluxesFK,σ(u) and the valuesuK , (uσ′)σ′∈EK

. But the idea is now, defining the space of the numerical fluxesYD

YD = ((FK,σ)σ∈EK)K∈M, FK,σ ∈ R. (2.47)

to connect an elementF ∈ YD with an elementu ∈ XD,0 by

〈F,G〉Y =∑

K∈M

∑

σ∈EK

(uK − uσ)GK,σ, ∀G ∈ YD, (2.48)

through a bilinar form〈·, ·〉Y onYD. This bilinear form allows to consider the matrixMK defined by

Mσσ′K = 〈1K,σ,1K,σ′〉Y ,

and then the bilinear form can be rewritten in the equivalent form

〈F, G〉Y =∑

K∈M

∑

σ∈EK

∑

σ′∈EK

Mσσ′K FK,σGK,σ′ .

The scheme is then expressed by the relations (2.8)-(2.9)-(2.10) in addition to (2.48). Hence we must now definea suitable bilinear form onYD (this problem is explored by the mixed finite element method or the mimetic finitedifference method [6]). Instead of defining general orthogonality properties in the spaceYD, we here consider anidea similar to (2.40). Let us define, in the spirit of (2.39),

m(K)ΛKvK(F ) = −∑

σ∈EK

FK,σ(xσ − xK), (2.49)

with

ΛK =1

m(K)

∫

K

Λ(x)dx.

18

Note that the definition〈F,G〉Y =

∑

K∈Mm(K)ΛKvK(F ) · vK(G)

leads to a non-invertible matrix. We then introduce the linear form

HK,σ(F ) = −FK,σ

m(σ)− ΛKvK(F ) · nK,σ, (2.50)

andvK,σ(F ) = vK(F ) +

√dHK,σ(F )Λ−1

K

xσ − xK

dK,σ. (2.51)

We may then definevDF as the piecewise constant function equal tovK,σ(F ) a.e. in the coneDK,σ with vertexxK and basisσ:

vDF (x) = vK,σ(F ) for a.e.x ∈ DK,σ, (2.52)

and we set, following the same idea as (2.37)

〈F,G〉Y =∫

Ω

Λ(x)vDF (x) · vDG(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 diagonalmatrixMK in the case of isotropic diffision and of meshes which satisfynK,σ = xσ−xK

dK,σ(triangular, rectangular,

orthogonal parallelepipedic meshes but unfortunately not general tetrahedric meshes), and yields the usual twopoint 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 whereΛ is constant in each control volumeK ∈ M. Note that, for aslightly different definition ofvD, we may find exactly a mixed finite element method. Indeed, let us define

vK,σ(F )(x) = vK(F ) + HK,σ(F )Λ−1K

x− xK

dK,σ, ∀x ∈ DK,σ, (2.54)

andvDF (x) = vK,σ(F )(x) for a.e.x ∈ DK,σ, ∀K ∈M, ∀σ ∈ EK . (2.55)

This definition ensures the property

ΛK vK,σ(F )(x) · nK,σ = −FK,σ

m(σ), ∀x ∈ σ, ∀σ ∈ EK , ∀K ∈M.

Therefore, under the condition (2.10), we get thatΛK vDF ∈ Hdiv(Ω) (it is easy to see the continuity of the normalcomponent on the boundary of each coneDK,σ). DefiningVD = ΛK vDF, F ∈ YD 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 canbe written findg ∈ VD andu ∈ HM(Ω) such that

∫

Ω

g(x) · Λ−1(x)g(x)dx = −∫

Ω

u(x)divg(x)dx, ∀g ∈ VD, (2.56)

and ∫

Ω

u(x)divg(x)dx =∫

Ω

u(x)f(x)dx, ∀u ∈ HM(Ω). (2.57)

19

2.2 Numerical results

We present some numerical results obtained with various choices ofA in the scheme (2.28),(2.12) with the flux(2.46), which we synthetize here for the sake of clarity:

Findu ∈ XD,A (that is(uK)K∈M, (uσ)σ∈Ehyb), such that:∑

K∈M

∑

σ∈EK

FK,σ(u)(vK − vσ) =∑

K∈MvK

∫

K

f(x)dx, for all v ∈ XD,A,

with FK,σ(u) =∑

σ′∈EK

Aσσ′K (uσ′ − uK),∀K ∈M, ∀σ ∈ EK .

(2.58)

Order of convergenceWe consider here the numerical resolution of Equation (2.1) supplemented by a homoge-neous 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 aconstant matrixΛ:

Λ =(

1.5 .5.5 1.5

), (2.59)

and choosef : Ω → R andf such that the exact solution to Problem (2.1) isu : Ω → R defined byu(x, y) =16x(1 − x)y(1 − y) for any (x, y) ∈ Ω. Note that in this case, the composite scheme is in fact the cell centredscheme, 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 anduniform square meshes.

Figure 2.2: Regular conforming coarse and fine triangular grids

For bothA = ∅ (hybrid scheme) andA = Eint (cell centred scheme), the order of convergence is close to 2 forthe unknownu and 1 for its gradient. Of course, the hybrid scheme is almost three times more costly in terms ofnumber of unknowns than the cell centred scheme for a given precision. However, the number of nonzero terms inthe matrix is, again for a given precision on the approximate solution, larger for the cell centred scheme than forthe hybrid scheme. Hence the number of unknowns is probably not a sufficient criterion for assessing the cost ofthe scheme.Results were also obtained in the case of uniform square or rectangular meshes. They show a better rate ofconvergence 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 arectangular grid of3n × 2n (resp.5n × 2n) on the first (resp. second side), wheren is the number of the mesh,n = 1, . . . 7. . Again, the order of convergence which we obtain is2 for u and around1.8 for the gradient. Wegive in Table 2.1 below the errors obtained in the discreteL2 norm foru and∇u for a nonconforming mesh and(in terms of number of unknowns) and for the rectangular4 × 6 and4 × 10 conforming rectangular meshes, forboth the hybrid and cell centred schemes. We show on Figure 2.3 the solutions for the corresponding grids (whichlooks very much the same for both schemes).

20

NU NM ε(u) ε(∇u)n Hyb Cent Hyb Cent Hyb Cent Hyb Cent

C1 130 48 874 488 1.28E-01 1.20E-01 1.64E-02 3.57E-02NC 182 64 1334 724 1.03E-01 9.43E-02 1.66E-02 3.69E-02C2 222 80 1542 864 7.61E-02 7.09E-02 9.18E-03 2.44E-02

Table 2.1: Error for the non conforming rectangular mesh, hybrid scheme (Hyb) and centred (Cent) schemes. Forboth schemes: NU is the number of unknowns in the resulting linear system, NM the number of non zero terms inthe matrix,ε(u) the discreteL2 norm of the error on the solution andε(∇u) the discreteL2 norm of the error onthe gradient. C1 and C2 are the two conforming meshes represented on the left and the right in Figure 2.3, and NCthe non conforming one represented in the middle.

Figure 2.3: The approximate solution for conforming and nonconforming meshes. Left: conforming8 × 6 mesh,center: non conforming4× 6, 4× 10 mesh, right: conforming10× 10.

Further detailed results on several problems and conforming, non conforming and distorted meshes may be foundin [23].The case of a highly heterogeneous tilted barrierWe now turn to a heterogeneous case. The domainΩ =]0, 1[×]0, 1[ is composed of 3 subdomains, which aredepicted in Figure 2.4:Ω1 = (x, y) ∈ Ω; ϕ1(x, y) < 0, with ϕ1(x, y) = y − δ(x− .5)− .475, Ω2 = (x, y) ∈Ω; ϕ1(x, y) > 0, ϕ2(x, y) < 0, with ϕ2(x, y) = ϕ1(x, y)− 0.05, Ω3 = (x, y) ∈ Ω; ϕ2(x, y) > 0, andδ = 0.2is the slope of the drain (see Figure 2.4). Dirichlet boundary conditions are imposed by setting the boundary valuesto those of the analytical solution given byu(x, y) = −ϕ1(x, y) on Ω1 ∪ Ω3 andu(x, y) = −ϕ1(x, y)/10−2 onΩ2.The permeability tensorΛ is heterogeneous and isotropic, given byΛ(x) = λ(x)Id, with λ(x) = 1 for a.e.x ∈ Ω1 ∪ Ω3 andλ(x) = 10−2 for a.e. x ∈ Ω2. Note that the isolines of the exact solution are parallel to theboundaries of the subdomain, and that the tangential component of the gradient is 0. We use the meshes depictedin figure 2.4. Mesh 3 (containing10× 25 control volumes) is obtained from Mesh 1 by the addition of two layersof very thin control volumes around each of the two lines of discontinuity ofΛ: because of the very low thicknessof these layers, equal to1/10000, 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 = 1analytical -0.2 0.2 1. -1.

centredmesh 1 210 2424 -1.17 1.17 3.51 -3.51mesh 2 1000 11904 -0.237 0.237 1.104 -1.104mesh 3 250 2904 -0.208 0.208 1.02 -1.02

compositemesh 1 239 2583 -0.2 0.2 1. -1.mesh 2 1020 12036 -0.2 0.2 1. -1.

hybridmesh 1 599 4311 -0.2 0.2 1. -1.mesh 2 2890 21138 -0.2 0.2 1. -1.

21

Ω1

Ω2

Ω3

Figure 2.4: Domain and meshes used for the tilted barrier test: mesh 1 (10× 21 center), mesh 2 (10× 100 right)

Note that the values of the numerical solution given by the hybrid and composite schemes are equal to those of theanalytical one (this holds under the only condition that the interfaces located on the linesϕi(x, y) = 0, i = 1, 2are not included inA, and that, for allσ ∈ A, all K ∈ M with βK

σ 6= 0 are included in the same subdomainΩi).Note that Mesh 3, which leads to acceptable results for the computation of the fluxes, is not well designed for sucha coupled problem, because of too small measures of control volumes. Hence the composite method on Mesh 1appears to be the most suitable method for this problem.Tilted barrierThe domainΩ =]0, 1[×]0, 1[ is composed of 3 subdomains:Ω1 = (x, y) ∈ Ω; ϕ1(x, y) < 0, with ϕ1(x, y) =y − δ(x − .5) − .475, Ω2 = (x, y) ∈ Ω; ϕ1(x, y) > 0, ϕ2(x, y) < 0, andϕ2(x, y) = ϕ1(x, y) − 0.05,Ω3 = (x, y) ∈ Ω; ϕ2(x, y) > 0, andδ = 0.2 is the slope of the drain. We impose, at the boundary of thedomain, the values of the analytical solution given byu(x, y) = −ϕ1(x, y) onΩ1, u(x, y) = −ϕ1(x, y)/10−2 onΩ2, u(x, y) = −ϕ2(x, y)− 0.05/10−2 onΩ2.The permeability tensorΛ is heterogeneous and isotropic, given byΛ(x) = λ(x)Id, with λ(x) = 1 for a.e.x ∈ Ω1, λ(x) = 10−2 for a.e.x ∈ Ω2 andλ(x) = 1 for a.e.Ω3. Mesh 3 (containing10 × 25 control volumes)is obtained from Mesh 1 by the addition of two layers of very thin control volumes around each of the two linesof discontinuity ofΛ: because of the very low thickness of these layers, equal to1/10000, the picture representingMesh 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 = 1analytical -0.2 0.2 1. -1.

Ebar = Eint (mesh 1) 210 2424 -1.17 1.17 3.51 -3.51Ebar = Eint (mesh 2) 1000 11904 -0.296 0.296 1.27 -1.27Ebar = Eint (mesh 3) 250 2904 -0.208 0.208 1.02 -1.02

∅ ( Ebar ( Eint (mesh 1) 239 2583 -0.2 0.2 1. -1.Ebar = ∅ (mesh 1) 599 4311 -0.2 0.2 1. -1.

Note that in the cases∅ ( Ebar ( Eint andEbar = ∅, the numerical solution is equal to the analytical one (thisholds under the only conditions that the interfaces located on the linesϕi(x, y) = 0, i = 1, 2 are not included inEbar, and that, for allσ ∈ Ebar, all K ∈ Nσ are included in the same subdomainΩi). Note that Mesh 3, whichleads 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 suitablemethod 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 theschemes provided in this chapter. Let us first introduce some notations related to the mesh. LetD = (M, E ,P) be

22

a discretization ofΩ in the sense of definition 1.2. The size of the discretizationD is defined by:

hD = supdiam(K),K ∈M,

and the regularity of the mesh by:

θD = max(

maxσ∈Eint,K,L∈Mσ

dK,σ

dL,σ, maxK∈M,σ∈EK

hK

dK,σ

). (2.60)

For a given setA ⊂ Eint and for a given family(βKσ ) K∈M

σ∈Eint

satisfying property (2.25), we introduce some measure

of the resulting regularity with

θD,A = max

(θD, max

K∈M,σ∈EK∩A

∑L∈M |βL

σ ||xL − xσ|2h2

K

). (2.61)

Definition 2.2 (Continuous, coercive, consistent and symmetric families of fluxes)Let F be a family of dis-cretizations in the sense of definition 1.2. ForD = (M, E ,P) ∈ F , K ∈ M andσ ∈ E , we denote byFDK,σ alinear mapping fromXD to R, and we denote byΦ = ((FDK,σ)K∈M

σ∈E)D∈F . We consider the bilinear form defined

by〈u, v〉F =

∑

K∈M

∑

σ∈EK

FDK,σ(u)(vK − vσ), ∀(u, v) ∈ X2D. (2.62)

The family of numerical fluxesΦ is said to be continuous if there existsM > 0 such that

〈u, v〉F ≤ M |u|X |v|X , ∀(u, v) ∈ X2D, ∀D = (M, E ,P) ∈ F . (2.63)

The family of numerical fluxesΦ is said to be coercive if there existsα > 0 such that

α|u|2X ≤ 〈u, u〉F , ∀u ∈ XD, ∀D = (M, E ,P) ∈ F . (2.64)

The family of numerical fluxesΦ is said to be consistent (with Problem (2.1)) if for any family(uD)D∈F satisfying:

• uD ∈ XD,0 for all D ∈ F ,

• there existsC > 0 with |uD|X ≤ C for all D ∈ F ,

• there existsu ∈ L2(Ω) with limhD→0

‖ΠMuD − u‖L2(Ω) = 0 (we get from Lemma 5.7 thatu ∈ H10 (Ω)),

then the following holds

limhD→0

〈uD, PDϕ〉F =∫

Ω

Λ(x)∇ϕ(x) · ∇u(x)dx, ∀ϕ ∈ C∞c (Ω). (2.65)

Finally the family of numerical fluxesΦ is said to be symmetric if

〈u, v〉F = 〈v, u〉F , ∀(u, v) ∈ X2D, ∀D = (M, E ,P) ∈ F .

We may now state the general convergence theorem.

Theorem 2.1 LetF be a family of discretizations in the sense of definition 1.2; for anyD ∈ F , letA ⊂ Eint and(βK

σ ) K∈Mσ∈Eint

satisfying property (2.25). We assume that there existsθ > 0 such thatθD,A ≤ θ, for all D ∈ F , where

θD,A is defined by (2.61). Let Φ = ((FDK,σ)K∈Mσ∈E

)D∈F be a continuous, coercive and symmetric and consistent

family of numerical fluxes in the sense of definition 2.2. Let(uD)D∈F be the family of functions solution to (2.28)for all D ∈ F . ThenΠMuD converges inL2(Ω) to the unique solutionu of (2.5)ashD → 0. Moreover∇DuDconverges to∇u in L2(Ω)d ashD → 0.

23

Proof. We letv = uD in (2.28), we apply the Cauchy-Schwarz inequality to the right hand side. We get

〈uD, uD〉D =∫

Ω

f(x)ΠMuD(x)dx ≤ ‖f‖L2(Ω)‖ΠMuD‖L2(Ω).

We apply the Sobolev inequality (5.13) withp = 2, which gives in this case‖ΠMuD‖L2(Ω) ≤ C1‖ΠMu‖1,2,M.Using (2.84) and the consistency of the familyΦ of fluxes, we then have

α|ΠMuD|2X ≤ C1‖f‖L2(Ω)|uD|X .

This leads to the inequality

‖u‖1,2,M ≤ |uD|X ≤ C1

α‖f‖L2(Ω). (2.66)

Thanks to lemma 5.7, we get the existence ofu ∈ H10 (Ω), and of a subfamily extracted fromF , such that

‖ΠMuD − u‖L2(Ω) tends to0 ashD → 0. For a givenϕ ∈ C∞c (Ω), let us takev = PD,Aϕ in (2.28) (recall thatPD,Aϕ ∈ XD,A). We get

〈uD, PD,Aϕ〉F =∫

Ω

f(x)PMϕ(x)dx.

Let us remark that, thanks to the continuity of the familyΦ of fluxes, we have

〈uD, PD,Aϕ− PDϕ〉F ≤ MC1

α‖f‖L2(Ω) |PD,Aϕ− PDϕ|X .

Thanks to (2.25) and (2.61), we get the existence ofCϕ, only depending onϕ such that, for allK ∈ M and allσ ∈ A ∩ EK ,

|∑

L∈MβL

σ ϕ(xL)− ϕ(xσ)| ≤∑

L∈M|βL

σ ||xL − xσ|2Cϕ ≤ θD,ACϕh2K . (2.67)

We can then deducelim

hD→0|PD,Aϕ− PDϕ|X = 0. (2.68)

Thanks to the properties of subfamily extracted fromF , we can apply the consistency hypothesis on the familyΦof fluxes, which gives

limhD→0

〈uD, PDϕ〉F =∫

Ω

Λ(x)∇ϕ(x) · ∇u(x)dx.

Gathering the two above results leads to

limhD→0

〈uD, PD,Aϕ〉F =∫

Ω

Λ(x)∇ϕ(x) · ∇u(x)dx,

which concludes the proof that∫

Ω

Λ(x)∇ϕ(x) · ∇u(x)dx =∫

Ω

f(x)ϕ(x)dx.

Therefore,u is the unique solution of (2.5), and we get that the whole family(uD)D∈F converges tou ashD → 0.

Let us now prove the second part of the theorem. Letϕ ∈ C∞c (Ω) be given (this function is devoted to approximateu in H1

0 (Ω)). Thanks to the Cauchy-Schwarz inequality, we have∫

Ω

|∇DuD(x)−∇u(x)|2dx ≤ 3 (TD1 + TD2 + T3)

where

TD1 =∫

Ω

|∇DuD(x)−∇DPDϕ(x)|2dx,

24

TD2 =∫

Ω

|∇DPDϕ(x)−∇ϕ(x)|2dx,

and

T3 =∫

Ω

|∇ϕ(x)−∇u(x)|2dx.

We have, thanks to Lemma 2.7,lim

hD→0TD2 = 0. (2.69)

We have, thanks to Lemma 2.5 and to the coercivity of the family of fluxes, that there existsC2 such that

‖∇Dv‖2L2(Ω)d ≤ C212|v|2X ≤ C2〈v, v〉F , ∀v ∈ XD,

with C2 = C212α . Takingv = uD − PDϕ, we have

TD1 ≤ C2(〈uD, uD〉F − 2〈uD, PDϕ〉F + 〈PDϕ,PDϕ〉F ).

Using the result of convergence proved foruD and the consistency of the family of fluxes, we get

limhD→0

〈uD, PDϕ〉F =∫

Ω

∇u(x) · Λ(x)∇ϕ(x)dx. (2.70)

The fact that|PDϕ|X remains bounded, results from the regularity ofϕ and the regularity hypotheses of the familyof discretizations. Hence we can use the consistency of the family of fluxes, which writes in this case

limhD→0

〈PDϕ, PDϕ〉F =∫

Ω

∇ϕ(x) · Λ(x)∇ϕ(x)dx. (2.71)

Remarking that passing to the limithD → 0 in (2.28) withv = uD provides that〈uD, uD〉F converges to∫Ω∇u ·

Λ∇udx, we get that

limhD→0

〈uD − PDϕ, uD − PDϕ〉F =∫

Ω

∇(u− ϕ) · Λ∇(u− ϕ)dx ≤ λ

∫

Ω

|∇u−∇ϕ|2dx,

which yields

lim suphD→0

TD1 ≤ C2λ

∫

Ω

|∇u−∇ϕ|2dx.

From the above results, we obtain that there existsC3, independent ofD, such that∫

Ω

|∇DuD(x)−∇u(x)|2dx ≤ C3

∫

Ω

|∇ϕ(x)−∇u(x)|2dx + TD4 ,

with (noting thatϕ is fixed)lim

hD→0TD4 = 0. (2.72)

Let ε > 0. We can chooseϕ such that∫Ω|∇ϕ(x) − ∇u(x)|2dx ≤ ε, and we can then choosehD small enough

such thatTD4 ≤ ε. This completes the proof that

limhD→0

∫

Ω

|∇DuD(x)−∇u(x)|2dx = 0 (2.73)

in the case of a general continuous, coercive, consistent and symmetric family of fluxes.¤Let us write an error estimate, in the particular case thatΛ = Id and that the solution of (2.5) is regular enough.

25

Theorem 2.2 (Error estimate, isotropic case)We consider the particular caseΛ = Id, and we assume that thesolutionu ∈ H1

0 (Ω) of (2.5) is in C2(Ω). LetD = (M, E ,P) be a discretization in the sense of definition 1.2, letA ⊂ Eint be given, letA = (βK

σ )σ∈A,K∈M be a family of real numbers such that (2.25)holds, and letθ ≥ θD,Abe given (see (2.61)). Let (FK,σ)K∈M,σ∈E be a family of linear mappings fromXD to R, such that there existsα > 0 with

α|u|2X ≤ 〈u, u〉F , ∀u ∈ XD, (2.74)

defining〈u, v〉F by (2.62). We denote by

Ru =

( ∑

K∈M

∑

σ∈EK

dK,σ

m(σ)

(FK,σ(PD,Au) +

∫

σ

∇u(x) · nK,σdγ(x))2

)1/2

. (2.75)

Then the solutionuD of (2.28)verifies that there existsC4, only depends onα and onθ, such that

‖ΠMuD − PMu‖L2(Ω) ≤ C4Ru, (2.76)

and verifies that there existsC5, only depending onα, θ andu such that

‖∇DuD −∇u‖L2(Ω)d ≤ C5 (Ru + hD) . (2.77)

Remark 2.11 The extension of Theorem 2.2 to the caseu ∈ H2(Ω) could be studied in the cased = 2 or d = 3.Such a study, which demands a rather longer and more technical proof, is not expected to provide more informationon the link between accuracy and the regularity of the mesh than the result presented here.

Proof. Let v ∈ XD, since−∆u = f , we get:

−∑

K∈MvK

∫

K

∆u(x)dx =∫

Ω

f(x)ΠMv(x)dx. (2.78)

Thanks to the following equality (recall thatu ∈ C2(Ω) and therefore∇u · nK,σ is defined on each edgeσ)

−∑

K∈MvK

∫

K

∆u(x)dx = −∑

K∈M

∑

σ∈EK

(vK − vσ)∫

σ

∇u(x) · nK,σdγ(x),

we get that

〈PD,Au, v〉F =∫

Ω

f(x)ΠMv(x)dx +∑

K∈M

∑

σ∈EK

(FDK,σ(PD,Au) +

∫

σ

∇u(x) · nK,σdγ(x))

(vK − vσ).

Takingv = PD,Au− uD ∈ XD,A in this latter equality and using (2.78) we get

〈v, v〉F =∑

K∈M

∑

σ∈EK

(FDK,σ(PD,Au) +

∫

σ

∇u(x) · nK,σdγ(x))

(vK − vσ),

which leads, using (2.74) and the Cauchy-Schwarz inequality, to

α|v|X ≤ Ru. (2.79)

Using (2.84) and the Sobolev inequality (5.13) withp = 2 provides the conclusion of (2.76). Let us now prove(2.77). We have

‖∇DuD −∇u‖L2(Ω)d ≤ ‖∇DuD −∇DPD,Au‖L2(Ω)d + ‖∇DPD,Au−∇u‖L2(Ω)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‖∇DPD,Au − ∇u‖L2(Ω)d ≤ C6hD is obtained thanks to Lemma 2.7 and using a similar inequality to (2.67),replacingϕ by u. ¤

26

2.3.1 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 dis-cretization ofΩ in the sense of Definition 1.4. LetΦ = ((FDK,σ)K∈M

σ∈E)D∈F be defined by (2.20). ThenΦ 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(uD)D∈F be afamily satisfying:

• uD ∈ XD,0 for all D ∈ F ,

• there existsC > 0 with |uD|X ≤ C for all D ∈ F ,

• there existsu ∈ L2(Ω) with limhD→0

‖ΠMuD − u‖L2(Ω) = 0 (we get from Lemma 5.7 thatu ∈ H10 (Ω)),

For anyD ∈ F , we define the functionGD ∈ LD(Ω)d (see Definition 1.3), by

GDK,σ = d

uσ − uK

dK,σnK,σ, ∀K ∈M, ∀σ ∈ EK . (2.80)

Let us show thatGD weakly converges to∇u in L2(Ω)d ash(M) → 0. We first prolongu andGD by 0 outsideΩ. Let us first remark that, thanks to the Cauchy-Schwarz inequality and usingm(DK,σ) = m(σ)dK,σ/d,

‖GD‖2L2(Rd)d = d∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(uσ − uK

dK,σ

)2

= d|uD|2X .

Since‖GD‖L2(Rd)d remains bounded, we can extract fromF a subfamily, again denotedF , such that there existsG ∈ L2(Rd)d andGD weakly converges inL2(Rd)d to G ash(M) → 0. Let us prove thatG = ∇u, whichimplies, by uniqueness of the limit, that the whole family converges to∇u ash(M) → 0.Let ϕ ∈ C1

c (Ω)d be given. Let us denote byϕD ∈ LD(Ω)d the element defined by the value1m(σ)

∫σ

ϕ(x)dγ(x),in DK,σ for all K ∈Mσ, again prolonged by0 outsideΩ. We then have thatϕD converges toϕ in L2(Rd)d. Weget, noticing that the termsuσ vanish for the interior edges since1m(σ)

∫σ

ϕ(x)dγ(x)·nK,σ+ 1m(σ)

∫σ

ϕ(x)dγ(x)·nL,σ = 0, that

∫

Rd

GD(x) ·ϕ(x)dx = −∑

K∈MuK

∑

σ∈EK

∫

σ

ϕ(x)dγ(x)nK,σ = −∫

Rd

uD(x)div(ϕ)(x)dx,

which implies, by passing to the limith(M) → 0,∫

Rd

G(x) ·ϕ(x)dx = −∫

Rd

u(x)div(ϕ)(x)dx.

This proves thatu ∈ H1(Rd). Sinceu = 0 outsideΩ, we get thatu ∈ H10 (Rd), and that

∫

Rd

G(x) ·ϕ(x)dx =∫

Rd

∇u(x) ·ϕ(x)dx,

which proves thatG = ∇u. For anyD ∈ F , we now define the functionHD ∈ LD(Ω)d (see Definition 1.3), by

HDK,σ =

ϕ(xσ)− ϕ(xK)dσ

nK,σ +∇ϕ(xK)− (∇ϕ(xK) · nK,σ)nK,σ, ∀K ∈M, ∀σ ∈ EK . (2.81)

27

ThenHD converges to∇ϕ in L∞(Ω)d ash(M) → 0. Indeed, we have, forK ∈ M andσ ∈ EK , using thechoice ofxσ for ∆−adapted pointed finite volume discretizations,

ϕ(xσ)− ϕ(xK) = dK,σ∇ϕ(xK) · nK,σ +d2

K,σ

2RK,σ,

with |RK,σ| is bounded by someL∞ norm of the second order partial derivatives ofϕ, denoted byC7. We considerh(M) small enough, such thatϕ vanishes on allK ∈M such thatEK ∩ Eext 6= ∅, hence forσ ∈ EK ∩ Eext,

HDK,σ = 0 = ∇ϕ(xK).

We then get

|HDK,σ −∇ϕ(xK)| ≤ dσ

2C7 ≤ h(M)C7,∀K ∈M, ∀σ ∈ EK .

Using that

〈uD, PDϕ〉X =∫

Ω

λD(x)GD(x) ·HD(x)dx,

in whichλD(x) denotes the piecewise function equal toλK in eachK ∈M, we get, by weak-strong convergence,

limh(M)→0

〈uD, PDϕ〉X =∫

Ω

λ(x)∇u(x) · ∇ϕ(x)dx.

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 withA = ∅.

2.3.2 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 anisotropicdiffusion]Under hypothesis (1.5) and (2.3), let F be a family of pointed∆−adapted polygonal finite volume space dis-cretization ofΩ in the sense of Definition 1.4. LetΦ = ((FDK,σ)K∈M

σ∈E)D∈F be defined by (2.33).

ThenΦ is a continuous, coercive and symmetric and consistent family of numerical fluxes in the sense of definition2.2.

Proof. Let us first show that this family is coercive. We apply (2.88) to(RK,σu)2, definingβK > 0 below. Weget

(RK,σu)2 ≥ αKdβK

1 + βK

(uσ − uK

dK,σ

)2

− αKdβK |∇Ku|2. (2.82)

Hence ∫

Ω

(Λ(x)∇Du(x) · ∇Du(x) + RDu(x)2

)dx

≥∑

K∈M

((λ− αKdβK)m(K)|∇Ku|2 + αK

βK

1 + βK

∑

σ∈EK

m(σ)dK,σ

(uσ − uK

dK,σ

)2)

ChoosingβK = λαKd and using2ab

a+b ≥ min(a, b) for a, b > 0, we get∫

Ω

(Λ(x)∇Du(x) · ∇Du(x) + RDu(x)2

)dx

≥ min(1dλ, α)

∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(uσ − uK

dK,σ

)2

= min(1dλ, α)|uD|2X .

Hence we get the coerciveness property. The consistency property is resulting from the weak and strong conver-gence properties (Lemmas 2.3 and 2.4) of∇Du defined by (2.30).¤Let us state and prove the needed results on the discrete gradient defined by (2.30).

28

Lemma 2.3 LetF be a family of discretizations in the sense of definition 1.4 such that there existsθ > 0 withθ ≥ θD for all D ∈ F . Let(uD)D∈F be a family of functions, such that:

• uD ∈ XD,Eint for all D ∈ F ,

• there existsC > 0 with |uD|X ≤ C for all D ∈ F ,

• there existsu ∈ L2(Ω) with limhD→0

‖ΠMuD − u‖L2(Ω) = 0,

Thenu ∈ H10 (Ω) and∇DuD weakly converge inL2(Ω)d to∇u ashD → 0, where the operator∇D is defined by

(2.30).

Proof. Let us prolongΠMuD and∇DuD by 0 outside ofΩ. Thanks to the Cauchy-Schwarz inequality, we easilyget that

‖∇DuD‖2L2(Ω)d ≤ d|uD|2X .

Hence, up to a subsequence, there exists some functionG ∈ L2(Rd)d such that∇DuD weakly converges inL2(Rd)d to G ashD → 0. Let us show thatG = ∇u. Let ψ ∈ C∞c (Rd)d be given. We assume thath(M) issmall enough for ensuring that the support ofψ does not intersect any boundary control volume. Let us considerthe termTD5 defined by

TD5 =∫

Rd

∇DuD(x) ·ψ(x)dx.

UsingxL − xK = dσnK,σ, we get thatTD5 = TD6 + TD7 , with

TD6 =12

∑σ∈Eint

Mσ=K,L

m(σ)(uL − uK)nK,σ · (ψK + ψL), with ψK =1

m(K)

∫

K

ψ(x)dx,

andTD7 =

∑

K∈M

∑σ∈EK∩EintMσ=K,L

m(σ)uσ − uK

dK,σ(xσ − xK) · (ψK −ψL).

We compareTD6 with TD8 defined by

TD8 =∑

σ∈EintMσ=K,L

m(σ)(uL − uK)nK,σ ·ψσ,

with

ψσ =1

m(σ)

∫

σ

ψ(x)dγ(x).

We get that

(TD6 − TD8 )2 ≤∑

σ∈EintMσ=K,L

m(σ)dσ

(uL − uK)2∑

K∈M

∑

σ∈EK

m(σ)dK,σ|ψK + ψL

2−ψσ|2,

which leads to limhD→0

(TD6 − TD8 ) = 0.

Since

TD8 = −∑

K∈M

∑

σ∈EK

m(σ)uKnK,σ ·ψσ = −∫

Rd

ΠMuD(x)divψ(x)dx,

we get that limhD→0

TD8 = − ∫Rd u(x)divψ(x)dx. Let us now turn to the study ofTD7 . We have, using the Cauchy-

Schwarz inequality and the regularity of the mesh

(TD7 )2 ≤ C8|u|2Xh(M)2,

29

whereC8 only depends onϕ andθ. Hencelim

hD→0TD7 = 0.

This proves that the functionG ∈ L2(Rd)d is a.e. equal to∇u in Rd. Sinceu = 0 outside ofΩ, we get thatu ∈ H1

0 (Ω), and the uniqueness of the limit implies that the whole family∇DuD weakly converges inL2(Rd)d to∇u ashD → 0.¤Let us now state some strong consistency property of the discrete gradient applied to the interpolation of a regularfunction.

Lemma 2.4 LetD be a discretization ofΩ in the sense of Definition 1.4, and letθ ≥ θD be given. Then, for anyfunctionϕ ∈ C2(Ω), there existsC9 only depending ond, θ andϕ such that:

‖∇DPDϕ−∇ϕ‖(L∞(Ω))d ≤ C9hD, (2.83)

where∇D is defined by (2.30).

Proof. Taking into account definition (2.30) and (2.39), we write, for anyK ∈M,

∇KPDϕ−∇ϕ(xK) =1

m(K)

∑

σ∈EK

m(σ)(

ϕ(xσ)− ϕ(xK)dK,σ

−∇ϕ(xK) · nK,σ

)(xσ − xK).

Thanks to ∣∣∣∣ϕ(xσ)− ϕ(xK)

dK,σ−∇ϕ(xK) · nK,σ

∣∣∣∣ ≤ C10h(M),

whereC10 only depends onϕ, we conclude the proof.¤We then get the convergence of the scheme and an error estimate, using Theorem 2.1 withA = Eint.

Convergence study in the case of general meshes

In this section, we denote byxσ the center of gravity of σ ∈ E .For allv ∈ XD, we denote byΠMv ∈ HM(Ω) the piecewise function fromΩ toR defined byΠMv(x) = vK fora.e.x ∈ K, for all K ∈M. Using the Cauchy-Schwarz inequality, we have for allσ ∈ Eint with Mσ = K, L,

(vK − vL)2

dσ≤ (vK − vσ)2

dK,σ+

(vσ − vL)2

dL,σ, ∀v ∈ XD,

which leads to the relation‖ΠMv‖21,2,M ≤ |v|2X , ∀v ∈ XD,0. (2.84)

The following lemma provides an equivalence property between theL2-norm of the discrete gradient, defined by(2.38)-(2.42) and the norm| · |X .

Lemma 2.5 LetD be a discretization ofΩ in the sense of Definition 1.3, and letθ ≥ θD be given (whereθD isdefined by (2.60)). Then there existsC11 > 0 andC12 > 0 only depending onθ andd such that:

C11|u|X ≤ ‖∇Du‖L2(Ω) ≤ C12|u|X , ∀u ∈ XD, (2.85)

where∇D is defined by (2.38)-(2.42).

Proof. By definition,

‖∇Du‖2L2(Ω)d =∑

K∈M

∑

σ∈EK

m(σ)dK,σ

d|∇K,σu|2.

30

From the definition (2.41), thanks to (2.39) and to the definition (2.38), we get that

∑

σ∈EK

m(σ)dK,σ

dRK,σu nK,σ = 0, ∀K ∈M. (2.86)

Therefore,

‖∇Du‖2L2(Ω)d =∑

K∈M

(m(K)|∇Ku|2 +

∑

σ∈EK

m(σ)dK,σ

d(RK,σu)2

). (2.87)

Let us now notice that the following inequality holds:

(a− b)2 ≥ λ

1 + λa2 − λb2, ∀a, b ∈ R, ∀λ > −1. (2.88)

We apply this inequality to(RK,σu)2 for someλ > 0 and obtain:

(RK,σu)2 ≥ λd

1 + λ

(uσ − uK

dK,σ

)2

− λd|∇Ku|2( |xσ − xK |

dK,σ

)2

. (2.89)

This leads to

∑

σ∈EK

m(σ)dK,σ

d(RK,σu)2 ≥ λ

1 + λ

∑

σ∈EK

m(σ)dK,σ

(uσ − uK

dK,σ

)2

− λm(K)d|∇Ku|2θ2.

Choosingλ as

λ =1

dθ2, (2.90)

we get that

‖∇Du‖2(L2(Ω))d ≥ λ

1 + λ|u|2X ,

which shows the left inequality of (2.85). Let us now prove the right inequality. Remark that, thanks to theassumption thatK is xK-star-shaped, the property

∑

σ∈EK

m(σ)dK,σ = d m(K), ∀K ∈M (2.91)

holds. On one hand, using the definition (2.38) of∇Ku and (2.91), the Cauchy–Schwarz inequality leads to:

|∇Ku|2 ≤ (1

m(K))2

∑

σ∈EK

m(σ)dK,σ

(uσ − uK)2∑

σ∈EK

m(σ)dK,σ =d

m(K)

∑

σ∈EK

m(σ)dK,σ

(uσ − uK)2. (2.92)

On the other hand, by definition (2.41), and thanks to the regularity of the mesh (2.95), we have:

(RK,σu)2 ≤ 2d

((uσ − uK

dK,σ)2 + |∇Ku|2|xσ − xK

dK,σ|2

)≤ 2d

((uσ − uK

dK,σ)2 + θ2|∇Ku|2

). (2.93)

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 LetF be a family of discretizations in the sense of definition 1.3 such that there existsθ > 0 withθ ≥ θD for all D ∈ F . Let(uD)D∈F be a family of functions, such that:

• uD ∈ XD,0 for all D ∈ F ,

31

• there existsC > 0 with |uD|X ≤ C for all D ∈ F ,

• there existsu ∈ L2(Ω) with limhD→0

‖ΠMuD − u‖L2(Ω) = 0,

Thenu ∈ H10 (Ω) and∇DuD weakly converge inL2(Ω)d to∇u ashD → 0, where the operator∇D is defined by

(2.38)-(2.42).

Remark 2.12 Note that the proof thatu ∈ H10 (Ω) also results from (2.84), which allows to apply Lemma 5.7 of

the appendix in the particular casep = 2.

Proof. Let us prolongΠMuD and∇DuD by 0 outside ofΩ. Thanks to Lemma 2.5, up to a subsequence, thereexists some functionG ∈ L2(Rd)d such that∇DuD weakly converges inL2(Rd)d to G ashD → 0. Let us showthatG = ∇u. Let ψ ∈ C∞c (Rd)d be given. Let us consider the termTD9 defined by

TD9 =∫

Rd

∇DuD(x) ·ψ(x)dx.

We get thatTD9 = TD10 + TD11, with

TD10 =∑

K∈M

∑

σ∈EK

m(σ)(uσ − uK)nK,σ ·ψK , with ψK =1

m(K)

∫

K

ψ(x)dx,

and

TD11 =∑

K∈M

∑

σ∈EK

RK,σu nK,σ ·∫

DK,σ

ψ(x)dx.

We compareTD10 with TD12 defined by

TD12 =∑

K∈M

∑

σ∈EK

m(σ)(uσ − uK)nK,σ ·ψσ,

with

ψσ =1

m(σ)

∫

σ

ψ(x)dγ(x).

We get that

(TD10 − TD12)2 ≤

∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(uσ − uK)2∑

K∈M

∑

σ∈EK

m(σ)dK,σ|ψK −ψσ|2,

which leads to limhD→0

(TD10 − TD12) = 0.

Since

TD12 = −∑

K∈M

∑

σ∈EK

m(σ)uKnK,σ ·ψσ = −∫

Rd

ΠMuD(x)divψ(x)dx,

we get that limhD→0

TD12 = − ∫Rd u(x)divψ(x)dx. Let us now turn to the study ofTD11. Noting again that (2.86)

holds, we have:

TD11 =∑

K∈M

∑

σ∈EK

RK,σu nK,σ ·∫

DK,σ

(ψ(x)−ψK)dx.

Sinceψ is a regular function, there existsCψ only depending onψ such that| ∫DK,σ

(ψ(x) − ψK)dx| ≤CψhD

m(σ)dK,σ

d. From (2.93) and the Cauchy-Schwarz inequality, we thus get:

limhD→0

TD11 = 0.

32

This proves that the functionG ∈ L2(Rd)d is a.e. equal to∇u in Rd. Sinceu = 0 outside ofΩ, we get thatu ∈ H1

0 (Ω), and the uniqueness of the limit implies that the whole family∇DuD weakly converges inL2(Rd)d to∇u ashD → 0.¤Let us now state some strong consistency property of the discrete gradient applied to the interpolation of a regularfunction.

Lemma 2.7 LetD be a discretization ofΩ in the sense of Definition 1.3, and letθ ≥ θD be given. Then, for anyfunctionϕ ∈ C2(Ω), there existsC13 only depending ond, θ andϕ such that:

‖∇DPDϕ−∇ϕ‖(L∞(Ω))d ≤ C13hD, (2.94)

where∇D is defined by (2.38)-(2.42).

Proof. Taking into account definition (2.42), and using definition (2.40), we write:

|∇K,σPDϕ−∇ϕ(xK)| ≤ |∇KPDϕ−∇ϕ(xK)|+ |RK,σPDϕ|From (2.38), we have, for anyK ∈M,

∇KPDϕ =1

m(K)

∑

σ∈EK

m(σ)(ϕ(xσ)− ϕ(xK))

=1

m(K)

∑

σ∈EK

m(σ)(∇ϕ(xK) · (xσ − xK) + h2

KρK,σ

)nK,σ,

where|ρK,σ| ≤ Cϕ with Cϕ only depending onϕ. Thanks to (2.39) and to the regularity of the mesh, we get:

|∇KPDϕ−∇ϕ(xK)| ≤ 1m(K)

∑

σ∈EK

m(σ)h2K |ρK,σ| ≤ hK d Cϕθ.

From this last inequality, using Definition 2.41, we get:

|RK,σPDϕ| =

√d

dK,σ|ϕ(xσ)− ϕ(xK)−∇KPDϕ · (xσ − xK)|

≤√

d

dK,σ

(h2

KρK,σ + h2K d Cϕθ

)

≤√

dθ(hKCϕ + hKdCϕθ),

which concludes the proof.¤

Lemma 2.8 LetF be a family of discretizations in the sense of definition 1.3. We assume that there existsθ > 0with

θD ≤ θ, ∀D ∈ F , (2.95)

whereθD is defined by (2.60). LetΦ = ((FDK,σ)K∈MDσ∈EK

)D∈F be the family of fluxes defined by (2.43)-(2.46). Then

the familyΦ is a continuous, coercive, consistent and symmetric family of numerical fluxes in the sense of definition2.2.

Proof. Since the family of fluxes is defined by (2.43)-(2.46), it satisfies (2.37), and therefore we have:

〈u, v〉F =∫

Ω

∇Du(x) · Λ(x)∇Dv(x)dx, ∀u, v ∈ XD.

Hence the property〈u, v〉F = 〈v, u〉F holds. The continuity and coercivity of the familyΦ result from Lemma2.5 and the properties ofΛ, which give:〈u, v〉F ≤ λ‖∇Du‖L2(Ω)‖∇Dv‖L2(Ω) and〈u, v〉F ≥ λ‖∇Du‖2L2(Ω) foranyu, v ∈ XD. The consistency results from the weak and strong convergence properties of lemmas 2.6 and 2.7,which give∇DuD → ∇u weakly inL2(Ω) and∇DPDϕ → ∇ϕ in L2(Ω) as the mesh size tends to 0.¤Let us apply the error estimate, in the particular case thatΛ = Id and that the solution of (2.5) is regular enough.

33

Theorem 2.3 (Error estimate, isotropic case)Under the hypotheses and notations of Theorem 2.2 and in theparticular case where(FK,σ)K∈M,σ∈E is defined by (2.43)-(2.46), there existsC14, only depending onα, θ andu, such that

Ru ≤ C14hD. (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 allv ∈ XD,

FK,σ(v) = −∑

σ′∈EK

(∇Kv + RK,σ′v nK,σ′) · m(σ′)dK,σ′

dyσ′σ,

with

yσ′σ =

m(σ)m(K)

nK,σ +

√d

dK,σ

(1− m(σ)

m(K)nK,σ · (xσ − xK)

)nK,σ if σ = σ′

m(σ)m(K)

nK,σ −√

d

dK,σ′m(K)m(σ)nK,σ · (xσ′ − xK)nK,σ′ otherwise.

Using (2.39), we get that ∑

σ′∈EK

m(σ′)dK,σ′

dyσ′σ = m(σ)nK,σ.

Since it is easy to see that there existsC15 ∈ R+ such that|RK,σ′PD,Au| ≤ C15hK , we then obtain that thereexists someC16 ∈ R+ with

∣∣∣∣FK,σ(PD,Au) +∫

σ

∇u(x) · nK,σdγ(x)∣∣∣∣ ≤ C16m(σ)hK .

This easily leads to the conclusion of (2.96).¤

2.3.3 The convergence proof in the case of the mixed-type schemes

Lemma 2.9 LetF be a family of discretizations in the sense of definition 1.3. We assume that there existsθ > 0such that (2.95)holds, whereθD is again defined by (2.60). LetΦ = ((FDK,σ)K∈MD

σ∈EK

)D∈F be the family of fluxes

defined by (2.48)-(2.49)-(2.53). Then the familyΦ is a continuous, coercive, consistent and symmetric family ofnumerical fluxes in the sense of definition 2.2.

Proof. Let us remark that we easily get

m(K)|vK(F )|2 ≤ C17

∑

σ∈EK

m(σ)dK,σ

(FK,σ

m(σ)

)2

.

Using the Cauchy-Schwarz inequality, we then obtain

∑

σ∈EK

m(σ)dK,σ|vK,σ(F )|2 ≤ C18

∑

σ∈EK

m(σ)dK,σ

(FK,σ

m(σ)

)2

.

Hence, lettingGK,σ = m(σ)dK,σ

(uK − uσ) in (2.48), we get

‖u‖2X ≤ (〈F, F 〉Y )1/2(〈G,G〉Y )1/2 ≤ (〈F, F 〉Y C18λ)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

∑

σ∈EK

m(σ)dK,σ

dHK,σ(F )Λ−1

K

xσ − xK

dK,σ= 0

34

holds, which allows to write

∫

Ω

|vDF (x)|2dx =∑

K∈M

(m(K)|vK(F )|2 +

∑

σ∈EK

m(σ)dK,σ

dHK,σ(F )2

∣∣∣∣Λ−1K

xσ − xK

dK,σ

∣∣∣∣2)

.

We then get, using again(2.88) in order to get a lower bound ofHK,σ(F )2, the existence ofC19 such that

m(K)|vK(F )|2 ≥ C19

∑

σ∈EK

m(σ)dK,σ

(FK,σ

m(σ)

)2

,

which yields the continuity of the family of fluxes (applying the Cauchy-Schwarz inequality to the right-hand sideof (2.48)). Note that we shall again get a strong convergence property forvD(F ) do∇u.¤

Chapter 3

Nonlinear convection - degeneratediffusion problems

3.1 Actual problems and schemes

Let Ω be a bounded open subset ofRd, (d ∈ N?) with boundary∂Ω and letT ∈ R∗+. One considers the followingproblem.

ut(x, t) + div(q f(u)

)(x, t)−∆ϕ(u)(x, t) = 0, for (x, t) ∈ Ω× (0, T ). (3.1)

The initial condition is formulated as follows:

u(x, 0) = u0(x) for x ∈ Ω. (3.2)

The boundary condition is the following nonhomogeneous Dirichlet condition:

u(x, t) = u(x, t), for (x, t) ∈ ∂Ω× (0, T ). (3.3)

This problem arises in different physical contexts. One of them is the problem of two phase flows in a porousmedium, such as the air-water flow of hydrological aquifers. In this case, (3.1)-(3.3) represents the conservationof the incompressible water phase, described by the water saturationu, submitted to convective flows (first orderspace termsq(x, t) f(u)) and capillary effects (∆ϕ(u)). The expressionq(x, t) f(u) for the convective term in(3.1) appears to be a particular case of the more general expressionF (u, x, t), but since it involves the same toolsas 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)onΩ,

(H2) u0 ∈ L∞(Ω) andu ∈ L∞(∂Ω×(0, T )), u being the trace of a function ofH1(Ω×(0, T ))∩L∞(Ω×(0, T ))(also denotedu); one setsuI = min(infessu0, infessu) anduS = max(supessu0, supessu),

(H3) ϕ is a nondecreasing Lipschitz-continuous function, with Lipschitz constantΦ, and one defines a functionζsuch thatζ ′ =

√ϕ′,

(H4) f ∈ C1(R,R), f ′ ≥ 0; one setsF = maxs∈[uI ,uS ] f′(s),

(H5) q is the restriction toΩ× (0, T ) of a function ofC1(Rd × R,Rd),

35

36

(H6) div(q(x, t)) = 0 for all (x, t) ∈ Rd×(0, T ), wherediv(q(x, t)) =d∑

i=1

∂qi

∂xi(x, t), (qi is thei-eth component

of q) and

q(x, t).n(x) = 0, for a.e.(x, t) ∈ ∂Ω× (0, T ), (3.4)

(for x ∈ ∂Ω, n(x) denotes the outward unit normal toΩ at pointx).

Remark 3.1 The functionf is assumed to be non decreasing in Hypothesis (H3) of Assumption 3.1 for the sakeof simplicity. In fact, the convergence analysis which we present here would also hold without this monotonicityassumption using for instance a flux splitting scheme for the treatment of the convective termqf(u).

Under Assumption 3.1, (3.1)-(3.3) does not have, in the general case, strong regular solutions. Because of the pres-ence of a non-linear convection term, the expected solution is an entropy weak solution in the sense of Definition3.2 given below.

3.1.1 Finite volume approximation

We may now define the finite volume discretization of (3.1)-(3.3). LetD be a∆−adapted pointed polygonal finitevolume space-time discretization ofΩ × (0, T ) in the sense of Definitions 1.4 and 1.7. The initial condition isdiscretized by:

u0K =

1m(K)

∫

K

u0(x)dx, ∀K ∈M. (3.5)

In order to introduce the finite volume scheme, we need to define:

un+1σ =

1δtn m(σ)

∫ tn+1

tn

∫

σ

u(x, t)dγ(x)dt, ∀σ ∈ Eext, ∀n ∈ [[0, N ]], (3.6)

qn+1K,σ =

1δtn

∫ tn+1

tn

∫

σ

q(x, t) · nK,σdγ(x)dt, ∀K ∈M, ∀σ ∈ EK ,∀n ∈ [[0, N ]]. (3.7)

For alln ∈ N, un+1 ∈ HM(Ω) andg ∈ C0(R), we introduce the notations

un+1K,σ = un+1

L , ∀K ∈M, ∀σ ∈ EK ∩ Eint, Mσ = K,Lun+1

K,σ = un+1σ , ∀K ∈M, ∀σ ∈ EK ∩ Eext, Mσ = K, (3.8)

andδn+1K,σ g(u) = g(un+1

K,σ )− g(un+1K ), ∀K ∈M, ∀σ ∈ EK . (3.9)

An implicit finite volume schemefor the discretization of (3.1)-(3.3) is given by the following set of nonlinearequations, the discrete unknowns of which areu = (un+1

K )K∈M,n∈[[0,N ]]:

un+1K − un

K

δtnm(K) +

∑

σ∈EK

[(qn+1

K,σ )+f(un+1K )− (qn+1

K,σ )−f(un+1K,σ )− m(σ)

dσδn+1K,σ ϕ(u)

]= 0,

∀K ∈M, ∀n ∈ [[0, N ]],(3.10)

where(qn+1K,σ )+ and(qn+1

K,σ )− denote the positive and negative parts ofqn+1K,σ (i.e. (qn+1

K,σ )+ = max(qn+1K,σ , 0) and

(qn+1K,σ )− = −min(qn+1

K,σ , 0)).

37

Remark 3.2 The upwind discretization of the fluxqf(u) in (3.10) uses the monotonicity off and should bereplaced 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 allK ∈M andn ∈ [[0, N ]],∑

σ∈EK

qn+1K,σ =

∑

σ∈EK

[(qn+1K,σ )+ − (qn+1

K,σ )−] = 0. This leads to

∑

σ∈EK

((qn+1

K,σ )+f(un+1K )− (qn+1

K,σ )−f(un+1K,σ )

)= −

∑

σ∈EK

(qn+1K,σ )−δn+1

K,σ f(u). (3.11)

This property will be used in the following.

In Section (3.2.2) we shall prove the existence (Lemma 3.1) and the uniqueness (Lemma 3.4) of the solutionu = (un+1

K )K∈M,n∈[[0,N ]] to (3.5)-(3.10). We may then define the approximate solution to (3.1)-(3.3) associatedto a∆−adapted pointed polygonal finite volume space-time discretization ofD of Ω× (0, T ) by:

Definition 3.1 LetD be a∆−adapted pointed polygonal finite volume space-time discretization ofΩ× (0, T ) inthe sense of Definitions 1.4 and 1.7. The approximate solution of (3.1)-(3.3) associated to the discretizationD isdefined almost everywhere inΩ× (0, T ) by:

uD(x, t) = un+1K , ∀x ∈ K, ∀t ∈ (tn, tn+1), ∀K ∈M, ∀n ∈ [[0, N ]], (3.12)

where(un+1K )K∈M,n∈[[0,N ]] is the unique solution to (3.5)-(3.10).

3.1.2 A numerical example

We finally present some numerical results which we obtained by implementing the scheme which was studiedabove in a prototype code.The domainΩ is the unit square(0, 1) × (0, 1). We define two subregionsΩ1 = (0.1, 0.3) × (0.4, 0.6) andΩ2 = (0.7, 0.9)× (0.4, 0.6). The initial data is given by0.5 in Ω\ (Ω1∪Ω2), 1 in Ω1 and0 in Ω2. It is representedon upper left corner of the figure below. The boundary value is the constant0.5.The functionϕ is defined byϕ(s) = 0 if s ∈ [0, 0.5] andϕ(s) = 0.2(s− 0.5) if s ∈ [0.5, 1], so that the diffusioneffect only takes place in the areas where the saturationu is greater than .5. The functionf is defined byf(s) = sand the fieldq is defined byq(x, y) = (10(x − x2)(1 − 2y),−10(y − y2)(1 − 2x)). Hence there is a linearrotating convective transport.

We define a coarse mesh of 14 admissible triangles on the unit square, from which we obtain a fine mesh of 12 600triangles by refining these 14 triangles uniformly 30 times. This fine mesh is used for the computations.

The figure below presents the obtained results at times0.000, 0.007, 0.028 and0.112. The black points correspondto the value1, the white ones to the value0, with a continuous scale of greys between these values. One observesthat the initial value0 is transported, only modified by the numerical diffusion due to the convective upstreamweighting, and that, on the contrary, the initial value1 is rapidly smoothed, due to the effect of the parabolic termwhich is active on the range[0.5, 1].

38

Computed solution at timet = 0 (initial condition),t = 0.007, t = 0.028 andt = 0.112.

3.2 Further analysis

3.2.1 Continuous definitions and main convergence result

Definition 3.2 (Entropy weak solution) Under Assumption 3.1, a functionu is said to be an entropy weak solu-tion to (3.1)-(3.3) if it verifies:

u ∈ L∞(Ω× (0, T )), (3.13)

ϕ(u)− ϕ(u) ∈ L2(0, T ; H10 (Ω)), (3.14)

andu satisfies the following Kruzkov entropy inequalities:∀ ψ ∈ D+(Ω× [0, T )), ∀κ ∈ R,

∫

Ω×(0,T )

|u(x, t)− κ| ψt(x, t)+(f(u(x, t)>κ)− f(u(x, t)⊥κ)) q(x, t) · ∇ψ(x, t)−∇|ϕ(u)(x, t)− ϕ(κ)| · ∇ψ(x, t)

dxdt +

∫

Ω

|u0(x)− κ|ψ(x, 0)dx ≥ 0,

(3.15)where one denotes bya>b the maximum value between two real valuesa andb, and bya⊥b their minimum valueand whereD+(Ω× [0, T )) = ψ ∈ C∞c (Ω× R,R+), ψ(·, T ) = 0 .

39

This notion has been introduced by several authors ([8], [29]), who proven the existence of such a solution inbounded domains. In [29], the proof of existence uses strong BV estimates in order to derive estimates in time andspace for the solution of the regularized problem obtained by adding a small diffusion term. In [8], the existenceof a weak solution is proven using semigroup theory (see [2]), and the uniqueness of the entropy weak solution isproven using techniques which have been introduced by S.N. Krushkov and extended by J. Carrillo.In the present study, thanks to condition (3.4), boundary conditions are entirely taken into account by (3.14) anddo not appear in the entropy inequality (3.15). For studies of the continuous problem, one can refer to [29], whichuses the classical Bardos-Leroux-Nedelec formulation [1], or [8] in the case of a homogeneous Dirichlet boundarycondition on∂Ω without condition (3.4).

Let us mention some related work in the case of infinite domains (Ω = Rd): In [3], the authors prove the existencein the caseΩ = Rd, regularizing the problem with the “general kinetic BGK” framework to yield estimates ontranslates of the approximate solutions. Continuity of the solution with respect to the data for a more generalequation was studied by Cockburn and Gripenberg [13], and convergence of the discretization with an implicitfinite volume scheme was recently studied by Ohlberger [31].

We shall deal here with the case of a bounded domain. The aim of the present work is then to prove the convergenceof approximate solutions obtained using a finite volume method with general unstructured meshes towards theentropy weak solution of (3.1)-(3.3) as the mesh size and time step tend to 0. We state this result in Theorem 3.1in Section 3.1.1, after presenting the finite volume scheme. Then in Section 3.2.2, the existence and uniquenessof the solution to the nonlinear set of equations resulting from the finite volume scheme is proven, along withsome properties of the discrete solutions. In Section 3.2.3 we show some compactness properties of the family ofapproximate solutions. We show in Section 3.2.4 that there exists some subsequence of sequences of approximatesolutions which tends to a so-called “entropy process solution”, and in Section 3.2.5 we prove the uniqueness ofthis entropy process solution, which allows us to conclude to the convergence of the scheme in Section 3.2.6. Wefinally give an example of numerical implementation in Section 3.1.2.

Theorem 3.1 (Convergence of the approximate solution towards the entropy weak solution)Letξ ∈ R, consider a family of∆−adapted pointed polygonal finite volume space-time discretization ofΩ×(0, T )in the sense of Definitions 1.4 and 1.7 such that, for allD in the family, one hasξ ≥ reg(D). For a given∆−adapted pointed polygonal finite volume space-time discretizationD of this family, letuD denote the associatedapproximate solution as defined in Definition 3.1. Then:

uD → u ∈ Lp(Ω× (0, T )) ash(D) → 0, ∀p ∈ [1, +∞),

whereu is the unique entropy weak solution to (3.1)-(3.3).

The proof of this convergence theorem will be concluded in Section 3.2.6 after we lay out the properties of thediscrete solution (sections 3.2.2 and 3.2.3), its convergence towards an “entropy process solution” (Section 3.2.4)and a uniqueness result on this entropy process solution (Section 3.2.5).

Remark 3.4 All the results of this chapter also hold for explicit schemes, under a convenient CFL condition onthe time step and mesh size.

3.2.2 Existence, uniqueness and discrete properties

We state here the properties and estimates which are satisfied by the scheme which we introduced in the previoussection and prove existence and uniqueness of the solution to this scheme. All the discrete properties which weaddress here correspond to natural estimates which are satisfied, at least formally, by regular continuous solutions.Let us first start by anL∞ estimate:

Lemma 3.1 (L∞ estimate) Under Assumption 3.1, letD be a∆−adapted pointed polygonal finite volume space-time discretization ofΩ× (0, T ) in the sense of Definitions 1.4 and 1.7 and let(un+1

K )K∈M,n∈[[0,N ]] be a solutionof scheme (3.5)-(3.10). Then

40

uI ≤ un+1K ≤ uS , ∀K ∈M, ∀n ∈ [[0, N ]].

Proof.Let uM = max

L∈M,m∈[[0,N ]]um+1

L and letn ∈ [[0, N ]] andK ∈M such thatun+1K = uM . Equations (3.10) and (3.11)

yield

uM = un+1K = un

K +δtn

m(K)

∑

σ∈EK

((qn+1

K,σ )−δn+1K,σ f(u) +

m(σ)dσ

δn+1K,σ ϕ(u)

). (3.16)

If one assumes thatuM ≥ max

σ∈Eext,m∈[[0,N ]]um+1

σ , using the monotonicity ofϕ (which impliesδn+1K,σ ϕ(u) ≤ 0) and that off (which

impliesδn+1K,σ f(u) ≤ 0, one getsuM ≤ un

K , and thereforeuM ≤ u0K .

This shows that

uM ≤ max( maxσ∈Eext,m∈[[0,N ]]

um+1σ , max

L∈Mu0

L),

yieldinguM ≤ uS . By the same method, one shows that minL∈M,m∈[[0,N ]]

um+1L ≥ uI . ¤

A corollary of Lemma 3.1 is the existence of a solution(un+1K )K∈M,n∈[[0,N ]] to (3.5)-(3.10). (Uniqueness is proven

in Lemma (3.4) below).

Corollary 3.1 (Existence of the solution to the scheme)Under Assumption 3.1, letD be a∆−adapted pointedpolygonal finite volume space-time discretization ofΩ× (0, T ) in the sense of Definitions 1.4 and 1.7. Then thereexists a solution(un+1

K )K∈M,n∈[[0,N ]] to the scheme (3.5)-(3.10).

The proof of this corollary is an adaptation of the technique which was used in [17] for the existence of the solutionto an implicit finite volume scheme for the discretization of a pure hyperbolic equation.The two following lemmas express the monotonicity of the scheme. Both are used to derive continuous entropyinequalities.

Lemma 3.2 (Regular convex discrete entropy inequalities)Under Assumption 3.1, letD be a∆−adapted pointed polygonal finite volume space-time discretization ofΩ ×(0, T ) in the sense of Definitions 1.4 and 1.7 and letu = (un+1

K )K∈M,n∈[[0,N ]] be a solution to (3.5)-(3.10).

Then, for allη ∈ C2(R,R), with η′′ ≥ 0, for all µ and ν in C1(R,R) with µ′ = η′(ϕ) and ν′ = η′(ϕ)f ′, forall K ∈ M, andn ∈ [[0, N ]], there exist(un+1

K,σ )σ∈EKwith un+1

K,σ ∈ (min(un+1K , un+1

K,σ ), max(un+1K , un+1

K,σ )) for allσ ∈ EK satisfying

µ(un+1K )− µ(un

K)δtn

m(K) +∑

σ∈EK

((qn+1

K,σ )+ν(un+1K )− (qn+1

K,σ )−ν(un+1K,σ )− m(σ)

dσδn+1K,σ η(u)

)

+12

∑

σ∈EK

m(σ)dσ

η′′(ϕ(un+1K,σ ))(δn+1

K,σ ϕ(u))2 ≤ 0(3.17)

Proof.In order to prove (3.17), one multiplies Equation (3.10) byη′(ϕ(un+1

K )).The convexity ofµ yields

41

m(K)un+1

K − unK

δtnη′(ϕ(un+1

K )) ≥ m(K)µ(un+1

K )− µ(un+1K )

δtn. (3.18)

Using the convexity ofν and Remark 3.3, one gets

−∑

σ∈EK

(qn+1K,σ )−δn+1

K,σ f(u)η′(ϕ(un+1K )) ≥ −

∑

σ∈EK

(qn+1K,σ )−δn+1

K,σ ν(u)

≥∑

σ∈EK

((qn+1

K,σ )+ν(un+1K )− (qn+1

K,σ )−ν(un+1K,σ )

).

The Taylor-Lagrange formula gives, for allσ ∈ EK , the existence ofun+1

K,σ ∈ (min(un+1K , un+1

K,σ ), max(un+1K , un+1

K,σ )) such that

−δn+1K,σ ϕ(u) η′(ϕ(un+1

K )) = −δn+1K,σ η(ϕ(u)) +

12η′′(ϕ(un+1

K,σ ))(δn+1K,σ ϕ(u))2.

Then collecting the previous inequalities gives Inequality (3.17) .¤

Lemma 3.3 (Kruzkov’s discrete entropy inequalities) Under Assumption 3.1, letD be a∆−adapted pointedpolygonal finite volume space-time discretization ofΩ × (0, T ) in the sense of Definitions 1.4 and 1.7 and letu = (un+1

K )K∈M,n∈[[0,N ]] be a solution of the scheme (3.5)-(3.10).

Then, for allκ ∈ R, K ∈M andn ∈ [[0, N ]],

|un+1K − κ| − |un

K − κ|δtn

m(K) +∑

σ∈EK

[(qn+1

K,σ )+|f(un+1K )− f(κ)|

−(qn+1K,σ )−|f(un+1

K,σ )− f(κ)|]

−∑

σ∈EK

m(σ)dσ

δn+1K,σ |ϕ(u)− ϕ(κ)| ≤ 0

(3.19)

Proof. In order to prove Kruzkov’s entropy inequalities, one follows [17]. Equation (3.10) is rewritten as

B(un+1K , un

K , (un+1L )L∈NK

, (un+1σ )σ∈EK,ext) = 0, (3.20)

whereB is nonincreasing with respect to each of its arguments exceptun+1K . Consequently,

B(un+1K , un

K>κ, (un+1L >κ)L∈NK , (un+1

σ >κ)σ∈EK,ext) ≤ 0. (3.21)

SinceB(κ, κ, (κ)L∈NK , (κ)σ∈EK,ext) = 0, one gets

B(κ, unK>κ, (un+1

L >κ)L∈NK , (un+1σ >κ)σ∈EK,ext) ≤ 0. (3.22)

Using the fact thatun+1K >κ = un+1

K or κ, (3.21) and (3.22) give

B(un+1K >κ, un

K>κ, (un+1L >κ)L∈NK

, (un+1σ >κ)σ∈EK,ext) ≤ 0. (3.23)

In the same way one obtains

B(un+1K ⊥κ, un

K⊥κ, (un+1L ⊥κ)L∈NK

, (un+1σ ⊥κ)σ∈EK,ext) ≥ 0. (3.24)

Substracting (3.24) from (3.23) and remarking that for any nondecreasing functiong and all real valuesa, b,g(a>b)− g(a⊥b) = |g(a)− g(b)| yields Inequality (3.19).¤Let us now prove the uniqueness of the solution to (3.5)-(3.10) and define the approximate solution.

42

Lemma 3.4 (Uniqueness of the approximate solution)Under Assumption 3.1, letD be a∆−adapted pointedpolygonal finite volume space-time discretization ofΩ× (0, T ) in the sense of Definitions 1.4 and 1.7. Then thereexists a unique solution(un+1

K )K∈M,n∈[[0,N ]] to (3.5)-(3.10).

Proof.The existence of(un+1

K )K∈M,n∈[[0,N ]] was established in Corollary 3.1. There only remains to prove the uniquenessof the solution. Let(un+1

K )K∈M,n∈[[0,N ]] and (vn+1K )K∈M,n∈[[0,N ]] (settingv0

K = u0K) be two solutions to the

scheme (3.5)-(3.10). Following the proof of Lemma 3.3, one gets, for allK ∈M and alln ∈ [[0, N ]],

B(un+1K >vn+1

K , unK>vn

K , (un+1L >vn+1

L )L∈NK, (un+1

σ )σ∈EK,ext) ≤ 0, (3.25)

and

B(un+1K ⊥vn+1

K , unK⊥vn

K , (un+1L ⊥vn+1

L )L∈NK, (un+1

σ )σ∈EK,ext) ≥ 0, (3.26)

which by substraction give

|un+1K − vn+1

K | − |unK − vn

K |δtn

m(K) +∑

σ∈EK

[(qn+1

K,σ )+|f(un+1K )− f(vn+1

K )|−(qn+1

K,σ )−|f(un+1K,σ )− f(vn+1

K,σ )|]

−∑

σ∈EK

m(σ)dσ

δn+1K,σ |ϕ(u)− ϕ(v)| ≤ 0.

(3.27)

For a givenn ∈ [[0, N ]], one sums (3.27) onK ∈ M and multiplies byδtn. All the exchange terms betweenneighbouring control volume disappear, and because of the sign of the boundary terms (forσ ∈ EK,ext, sinceun+1

K,σ = vn+1K,σ , we getδn+1

K,σ |ϕ(u)− ϕ(v)| ≤ 0), one gets

∑

K∈M|un+1

K − vn+1K |m(K) ≤

∑

K∈M|un

K − vnK |m(K). (3.28)

Sinceu0K = v0

K , one concludes∑

K∈M|un+1

K − vn+1K |m(K) = 0, for all n ∈ [[0, N ]], which concludes the proof of

uniqueness.¤

Let us now give two discrete estimates on the approximate solutionuD which will be crucial in the convergenceanalysis. The first estimate (3.32) is a discreteL2(0, T, H1(Ω)) estimate on the functionζ(uD) whereζ ′ =

√ϕ′.

This estimate will yield some compactness onζ(uD).The second estimate is the weakBV inequality (3.32) onf(uD). Such an inequality also holds for the continuousproblem with an additional diffusion term−ε∆f(u). This inequality does not give any compactness property (toour knowledge, noBV estimate is known in the case of unstructured meshes); however it it plays an essentialrole in the proof of convergence, where it is used to control the numerical diffusion introduced by the upstreamweighting scheme (see Section 3.2.4 and references [10], [12], [17] and [9]. Let us introduce a measure for theregularity of the scheme.

reg(D) = max(

maxσ∈Eint,K,L∈Mσ

dK,σ

dL,σ, maxK∈M,σ∈EK

hK

dK,σ

). (3.29)

Lemma 3.5 (DiscreteH1 estimate and weakBV inequality)Under Assumption 3.1, letD be a∆−adapted pointed polygonal finite volume space-time discretization ofΩ ×(0, T ) in the sense of Definitions 1.4 and 1.7. Letξ ∈ R be such thatξ ≥ reg(D); let (un+1

K )K∈M,n∈[[0,N ]] be thesolution of the scheme (3.5)-(3.10).

43

For all n ∈ N, un+1 ∈ HM(Ω) andg ∈ C0(R), we introduce the notation

δn+1σ g(u) = |g(uL)− g(uK)|, ∀σ ∈ Eint, Mσ = K, L,

δn+1σ g(u) = |g(un+1

σ )− g(uK)|, ∀σ ∈ Eext, Mσ = K. (3.30)

We also define

qn+1σ =

∣∣∣ 1δtn

∫ tn+1

tn

∫σq(x, t) · nK,σdγ(x)dt

∣∣∣ , ∀σ ∈ Eint, Mσ = K, L. (3.31)

Then there exists a real numberC > 0, only depending onΩ, T, u0, u, f,q, ϕ andξ such that

(ND(ζ(uD)))2 =N∑

n=0

δtn∑

σ∈E

m(σ)dσ

(δn+1σ ζ(u))2 ≤ C

(BD(f(uD)))2 =N∑

n=0

δtn∑

σ∈Eint

qn+1σ (δn+1

K,σ f(u))2 ≤ C (3.32)

Proof. One first defines discrete values by averaging, in each control volume, the functionu, whose traceon ∂Ω defines the Dirichlet boundary condition. Note that this proof usesu ∈ H1(Ω × (0, T )) and not onlyu ∈ L2(0, T ; H1(Ω)) andut ∈ L2(0, T ; H−1(Ω)), since we use below the fact thatut ∈ L2(0, T ;L1(Ω)). Let

u0K =

1m(K)

∫

K

u(x, 0)dx, ∀K ∈M, (3.33)

un+1K =

1δtn m(K)

∫ tn+1

tn

∫

K

u(x, t)dxdt, ∀K ∈M, ∀n ∈ [[0, N ]], (3.34)

Settingv = u − u, one multiplies (3.10) byδtnvn+1K and sums overK ∈ M andn ∈ [[0, N ]]. This yields

T1 + T2 + T3 = 0 with

T1 =N∑

n=0

∑

K∈Mm(K)(un+1

K − unK)vn+1

K , (3.35)

T2 =N∑

n=0

δtn∑

K∈M

∑

σ∈EK

((qn+1K,σ )+f(un+1

K )− (qn+1K,σ )−f(un+1

K,σ ))vn+1K , (3.36)

T3 =N∑

n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dσ

δn+1K,σ ϕ(u) vn+1

K . (3.37)

Usingu = v + u yieldsT1 = T4 + T5 with

T4 =12

∑

K∈Mm(K)((vN+1

K )2 − (v0K)2) +

12

N∑n=0

∑

K∈Mm(K)(vn+1

K − vnK)2 (3.38)

T5 =N∑

n=0

∑

K∈Mm(K)(un+1

K − unK)vn+1

K . (3.39)

Setting

44

An,K = un+1K − 1

m(K)

∫

K

u(x, tn)dx andBn,K =1

m(K)

∫

K

u(x, tn)− unK ,

one has

T5 =N∑

n=0

∑

K∈Mm(K)An,Kvn+1

K +N∑

n=0

∑

K∈Mm(K)Bn,Kvn+1

K .

By a classical density argument one gets:

|An,K | ≤ 1m(K)

‖ut‖L1(K×(tn,tn+1)), ∀n ∈ [[0, N ]], ∀K ∈M

and

|Bn,K | ≤ 1m(K)

‖ut‖L1(K×(tn−1,tn)), ∀n ∈ [[1, N ]], ∀K ∈M

(note thatB0,K = 0 for all K ∈ M). Using these two inequalities and theL∞ stability of the scheme (Lemma3.1) yields:

|T5| ≤ 2‖ut‖L1(Ω×(0,T ))(uS − uI).

Now remarking that

T4 ≥ −12

∑

K∈Mm(K)v0

K2 ≥ −1

2‖u0 − u(·, 0)‖2L2(Ω)

the previous inequality allows us to obtain the existence ofT6 > 0, only depending onΩ, T, u0 andu, such thatT1 ≥ T6 .

The termT2 can be decomposed inT2 = T7 + T8 with

T7 =N∑

n=0

δtn∑

K∈M

∑

σ∈EK

((qn+1K,σ )+f(un+1

K )− (qn+1K,σ )−f(un+1

K,σ ))un+1K ,

T8 = −N∑

n=0

δtn∑

K∈M

∑

σ∈EK

((qn+1K,σ )+f(un+1

K )− (qn+1K,σ )−f(un+1

K,σ )) un+1K ,

Using Remark 3.3, one gets

T7 =N∑

n=0

δtn∑

K∈M

∑

σ∈EK

(qn+1K,σ )−δn+1

K,σ f(u) un+1K . (3.40)

Let g be a primitive off andg(s) = sf(s)− g(s) for all reals. The following inequality holds for all pairs of realvalues(a, b) (see [20] and [9]).

g(b)− g(a) ≤ b(f(b)− f(a))− 12F

(f(b)− f(a))2 (3.41)

Using (3.41) for(a, b) = (un+1L , un+1

K ) and (3.40) yield

45

T7 ≥N∑

n=0

δtn∑

K∈M

∑

σ∈EK

(qn+1K,L )−δn+1

K,σ g(u) +1

2F(BD(f(uD)))2.

Using Remark 3.3 withg instead off gives

N∑n=0

δtn∑

K∈M

∑

σ∈EK

(qn+1K,L )−δn+1

K,σ g(u) = 0, (3.42)

and therefore

T7 ≥ 12F

(BD(f(uD)))2. (3.43)

A discrete space integration by parts inT8 does not yield any boundary term sinceq · n = 0 on ∂Ω, and gives,using the Cauchy-Schwarz inequality,

T8 = −N∑

n=0

δtn∑

K|L∈Eint

((qn+1K,L )+f(un+1

K )− (qn+1K,L )−f(un+1

L ))(un+1K − un+1

L )

≥ −‖q‖L∞(Ω×(0,T )) maxs∈[uI ,uS ]

|f(s)|N∑

n=0

δtn∑

σ∈Eint

m(σ)δn+1σ u

≥ −‖q‖L∞(Ω×(0,T )) maxs∈[uI ,uS ]

|f(s)|ND(uD)

(N∑

n=0

δtn∑

σ∈Eint

m(σ)dσ

) 12

≥ −ND(uD)‖q‖L∞(Ω×(0,T )) maxs∈[uI ,uS ]

|f(s)|(d m(Ω) T )12 .

The following estimate forND(uD) holds:

ND(uD) ≤ F (ξ)‖u‖L2(0,T,H1(Ω)), (3.44)

whereF ≥ 0 only depends onξ (Inequality (3.44) is proven in [19], with a different definition of the regularityfactor of the mesh), leading to a lower bound ofT8 denoted byC1, only depending onΩ, T, u0, u, f,q andξ.

There only remains to deal withT3. A discrete space integration by parts, using the fact thatV n+1

σ = 0, ∀σ ∈ Eext, ∀n ∈ [[0, N ]], yields

T3 =N∑

n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dK,σ

δn+1K,σ ϕ(u)

dσ

δn+1K,σ v

dσ. (3.45)

Writing againv into u− u leads toT3 = T9 + T10 where

T9 =N∑

n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dK,σ

δn+1K,σ ϕ(u)

dσ

δn+1K,σ u

dσ(3.46)

T10 = −N∑

n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dK,σ

δn+1K,σ ϕ(u)

dσ

δn+1K,σ u

dσ(3.47)

46

One has for all pairs of real numbers(a, b) the inequality(ζ(a) − ζ(b))2 ≤ (a − b)(ϕ(a) − ϕ(b)). Also usingϕ′ ≤ √

Φζ ′ (recall thatΦ = ‖ϕ′‖∞), one gets

T9 ≥ (ND(ζ(uD)))2, (3.48)

T10 ≥ −√

ΦND(ζ(uD))ND(uD). (3.49)

Using the Young inequality and (3.44), one gets the existence ofC2 only depending onΩ, T, u0, u, f,q, ϕ andξsuch that

T10 ≥ −12(ND(ζ(uD)))2 + C2. (3.50)

Gathering the previous inequalities, one gets

T6 +1

2F(BD(f(uD)))2 + T1 +

12(ND(ζ(uD)))2 + C2 ≤ 0, (3.51)

which completes the proof.¤

Remarking that from the estimate of Lemma 2 in [19], one hasND(ζ(uD)) ≤ √ΦC‖u‖L2(0,T,H1(Ω)), where

C ≥ 0 only depends onξ, one gets

Corollary 3.2 (DiscreteH10 estimate) Under Assumption 3.1, letD be a∆−adapted pointed polygonal finite

volume space-time discretization ofΩ× (0, T ) in the sense of Definitions 1.4 and 1.7. Letξ ∈ R be such thatξ ≥reg(D), let u = (un+1

K )K∈M,n∈[[0,N ]] be the solution of the scheme (3.5)-(3.10)and letu = (un+1K )K∈M,n∈[[0,N ]]

be defined by (3.34). Then, settingz = ζ(u)− ζ(u), there existsC ′ ∈ R+, only depending onΩ, T, u0, u, ϕ,q, fandξ such that

N∑n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(δn+1K,σ z

dσ

)2

=N∑

n=0

δtn∑

σ∈E

m(σ)dσ

(δn+1σ z

)2 ≤ C ′ (3.52)

3.2.3 Compactness of a family of approximate solutions

From Lemma 3.1, we know that for any sequence of∆−adapted pointed polygonal finite volume space-time dis-cretizations(Dm)m∈N, of Ω×(0, T ) in the sense of Definitions 1.4 and 1.7, the associated sequence of approximatesolutions(uDm)m∈N is bounded inL∞(Ω × (0, T )). Therefore one may extract a subsequence which convergesfor the weak star topology ofL∞(Ω× (0, T )) asm tends to infinity. This convergence is unfortunately insufficientto pass to the limit in the nonlinearities. In order to pass to the limit, we shall use two tools:

1. the nonlinear weak star convergence which was introduced in [17] and which is equivalent to the notion ofconvergence towards a Young measure as developped in [15].

2. Kolmogorov’s compactness theorem, which was used in [19] in the case of a semilinear elliptic equation.

Let us now show that we are in position to apply the Kolmogorov’s compactness theorem to(ζ(uDm))m∈N. Fromthe discrete estimates Lemma 3.5 and Corollary 3.2, one can state the following continuous estimates onzD, wherezD is defined almost everywhere inΩ× (0, T ) by

zD(x, t) = ζ(un+1K )− ζ(un+1

K ) for x ∈ K andt ∈ (tn, tn+1) (3.53)

where(un+1K )K∈M,n∈[[0,N ]] is the solution to (3.5)-(3.10) and(un+1

K )K∈M,n∈[[0,N ]] is defined by (3.34).

47

Corollary 3.3 (Space and time translates estimates)Under Assumption 3.1, letD be a ∆−adapted pointedpolygonal finite volume space-time discretization ofΩ × (0, T ) in the sense of Definitions 1.4 and 1.7. Letξbe a real number such thatξ ≥ reg(D); let u be the solution of scheme (3.5)-(3.10), and letuD be defined by(3.12). Let u be defined by (3.34), let zD be defined by (3.53), and be prolonged by zero on(0, T )×Ωc. Then thereexistsC1 only depending onΩ, T, u0, u, ϕ,q, f andξ, and there existsC0, only depending onΩ, such that

∀ξ ∈ Rd,

∫ T

0

∫

Rd

(zD(x + ξ, t)− zD(x, t))2dxdt ≤ C1|ξ|(|ξ|+ C0 h(M)), (3.54)

and there existsC2 only depending onΩ, T, u0, u, ϕ,q, f andξ such that

∀s > 0,

∫ T−s

0

∫

Rd

(ζ(uD)(x, t + s)− ζ(uD)(x, t))2dxdt ≤ C2 s. (3.55)

The use of space translate estimates for the study of numerical schemes for elliptic problems was recently intro-duced in [19]. The technique of [19] may easily be adapted here to prove (3.54), using the estimates of Corollary3.2. A time translate estimate was introduced in [16] to obtain some compactness in the study of finite volumeschemes for parabolic equations. The proof of (3.55) follows the technique of [16] and uses estimate (3.32) andthe discrete equation (3.10).

From Kolmogorov Copmpactness Theorem and the estimates (3.54) and (3.55) of Corollary 3.3 we deduce thefollowing compactness result:

Corollary 3.4 (Compactness of a family of approximate solutions)Let(Dm)m∈N be a sequence of∆−adapted pointed polygonal finite volume space-time discretization ofΩ×(0, T )in the sense of Definitions 1.4 and 1.7 such that there existsξ ∈ Rwith ξ ≥ reg(Dm) for all m ∈ N. For all m ∈ N,let uDm be defined by the scheme (3.5)-(3.10)and (3.12)with D = Dm, and letzDm be defined by (3.53)withD = Dm and (3.34). Then there existsu ∈ L∞(Ω× (0, T )× (0, 1)) andz ∈ L2(Ω× (0, T )) such that, up to asubsequence,uDm tends tou in the nonlinear weak star sense andzDm tends toz in L2(Ω× (0, T )) asm →∞.Furthermore one hasz ∈ L2(Ω× (0, T ))(0, T,H1

0 (Ω)), ζ(u) = z + ζ(u), andζ(u) = ζ(u) a.e. on∂Ω.

Proof. The convergence ofuDm towardsu ∈ L∞(Ω× (0, T )× (0, 1)) in the nonlinear weak star sense is aconsequence of Lemma 3.1 and Theorem 3.2, stated at the end of the present proof. The convergence ofzDm to zin L2(Ω × (0, T ) is a consequence of Kolmogorov’s compactness theorem and the estimates (3.54) and (3.55) ofCorollary 3.3.

Following [20] or [19], one then deduces from (3.55) thatDiz ∈ L2(Ω × (0, T )) for i = 1, . . . , d and sincezDm(x, t) = 0 onΩc × (0, T ) for all m ∈ N, one hasz ∈ L2(Ω× (0, T ))(0, T,H1

0 (Ω)).Now sinceuDm converges tou in the nonlinear weak star sense and that the functionuDm defined a.e. by

uDm(x, t) = un+1K for (x, t) in K × (tn, tn+1) converges uniformly tou, one deduces thatζ(uDm) converges

to ζ(u) in the nonlinear weak star sense and toz + ζ(u) in L2(Ω × (0, T )) asm tends to infinity. Therefore,by Lemma 3.6 below, one obtains thatζ(u) = z + ζ(u) andζ(u) does not depend onα. Furthermore, sincez ∈ L2(Ω× (0, T ))(0, T,H1

0 (Ω)), it follows thatζ(u) = ζ(u) a.e. on∂Ω which ends the proof of the corollary.¤

Theorem 3.2 (Nonlinear weak star convergence)Let Q be a Borelian subset ofRk, k ∈ N?, and(un)n∈N be abounded sequence inL∞(Ω × (0, T ))(Q). Then there existsu ∈ L∞(Ω × (0, T ))(Q × (0, 1)), such that up to asubsequence,un tends tou “in the nonlinear weak star sense” asn →∞, i.e.:

∀g ∈ C(R,R), g(un)

∫ 1

0

g(u(·, α))dα for the weak star topology ofL∞(Ω× (0, T ))(Q) asn →∞. (3.56)

We refer to [15; 17] for details and proof of Theorem 3.2.

48

Lemma 3.6 Let Q be a Borelian subset ofRk, k ∈ N?, and let(un)n∈N ⊂ L∞(Q) be such thatun converges tou ∈ L∞(Ω× (0, T ))(Q× (0, 1)) in the nonlinear weak star sense, and tow in L2(Q), asn tends to infinity, thenu(x, α) = w(x), for a.e.(x, α) ∈ Q× (0, 1) andu does not depend onα.

Proof. With the notations of the lemma, we have

∫ 1

0

∫

Q

(u(x, α)−w(x))2dxdα =∫ 1

0

∫

Q

(u(x, α))2dxdα−2∫ 1

0

∫

Q

u(x, α)w(x)dxdα+∫ 1

0

∫

Q

w(x)2dxdα.

Sinceun tends tou in the nonlinear weak star sense , one has

∫ 1

0

∫

Q

(u(x, α))2dxdα = limn→+∞

∫

Q

(un(x))2dx and∫ 1

0

∫

Q

u(x, α)w(x)dxdα = limn→+∞

∫

Q

un(x)w(x)dx,

and sinceun tends tow in L2(Q), one deduces thatu(x, α) = w(x), for a.e.(x, α) ∈ Q× (0, 1) andu does notdepend onα. ¤

3.2.4 Convergence towards an entropy process solution

This section is mainly devoted to the proof of the convergence theorem 3.3, which states the convergence of theapproximate solution to a measure valued solution as introduced in [15], which is also called entropy processsolution [17], and defined as follows.

Definition 3.3 Under Assumption 3.1, an entropy process solution to (3.1)-(3.3) is a functionu such that,

u ∈ L∞(Ω× (0, T )× (0, 1)), (3.57)

ϕ(u)− ϕ(u) ∈ L2(0, T ; H10 (Ω)), (3.58)

(note thatϕ(u) does not depend onα), andu satisfies the following inequalities :

1. Regular convex entropy inequalities :

∫

Ω×(0,T )

∫ 1

0µ(u(x, t, α))dα ψt(x, t)+∫ 1

0ν(u(x, t, α))dα q(x, t) · ∇ψ(x, t)

−∇η(ϕ(u)(x, t)) · ∇ψ(x, t)−η′′(ϕ(u)(x, t))(∇ϕ(u)(x, t))2ψ(x, t)

dxdt +

∫

Ω

µ(u0(x))ψ(x, 0)dx ≥ 0,

∀ ψ ∈ D+(Ω× [0, T )), ∀η ∈ C2(R,R), η′′ ≥ 0, µ′ = η′(ϕ(·)), ν′ = η′(ϕ(·))f ′(·).

(3.59)

2. Kruzkov’s entropy inequalities :

∫

Ω×(0,T )

∫ 1

0|u(x, t, α)− κ|dα ψt(x, t)+∫ 1

0(f(u(x, t, α)>κ)− f(u(x, t, α)⊥κ))dα q(x, t) · ∇ψ(x, t)

−∇|ϕ(u)(x, t)− ϕ(κ)| · ∇ψ(x, t)

dxdt

+∫

Ω

|u0(x)− κ|ψ(x, 0)dx ≥ 0,

∀ ψ ∈ D+(Ω× [0, T )), ∀κ ∈ R.

In the previous definition, we use two types of entropies, since in the proof (given below) of the uniqueness theoremone should make use of termsη′′(ϕ(u)). In [8], these terms are obtained from the equation satisfied by a weaksolution, which itself can be obtained from the Krushkov entropy inequalities. We have prefered here to keep thisslightly more complex definition since the following theorem shows that (3.59) and (3.60) are both obtained by thenatural limit of the approximate solutions.

49

Theorem 3.3 (Convergence towards an entropy process solution)Under Assumption 3.1, let(Dm)m∈N be asequence of∆−adapted pointed polygonal finite volume space-time discretization ofΩ × (0, T ) in the sense ofDefinitions 1.4 and 1.7, withh(Dm) → 0 as m → ∞, such that there existsξ ∈ R with ξ ≥ reg(Dm) for allm ∈ N. For all m ∈ N, let uDm be defined by the scheme (3.5)-(3.10)and (3.12)withD = Dm.Then, there exists an entropy processus solution of (3.1)-(3.3) in the sense of Definition 3.3 and a subsequence of(uDm)m∈N, again denoted by(uDm)m∈N, such that(uDm)m∈N converges tou in the nonlinear weak star senseand(ζ(uDm

))m∈N converges inL2(Ω× (0, T )) to ζ(u) ∈ L2(0, T ;H1(Ω)) asm tends to∞.

Proof. By Lemma 3.4, there existu ∈ L∞(Ω× (0, T )× (0, 1)) and a subsequence of(uDm)m∈N, again denoted

(uDm)m∈N, such that(uDm)m∈N converges tou in the nonlinear weak star sense and(ζ(uDm))m∈N converges inL2(Ω× (0, T )) to ζ(u) ∈ L2(0, T ; H1(Ω)). There remains to show that the functionu ∈ L∞(Ω× (0, T )× (0, 1))is an entropy process solution.A number of the arguments involved in order to do so may be found in [17] or [16] and therefore will be givenwith few details. The main new argument introduced here concerns the term∫

Ω×(0,T )

η′′(ϕ(u)(x, t))(∇ϕ(u)(x, t))2ψ(x, t)dxdt in equation (3.59). The passage to the limit to obtain this

nonlinearity motivates the use of Lemma 5.13 (a related technique was used in [27] in the case of a variationalinequality).

The idea of the proof is to derive the continuous inequalities (3.59) and (3.60) for the limitu by mutliplyingthe discrete entropy inequalities (3.17) and (3.19) by regular test functions and passing to the limit. Indeed, letψ ∈ D+(Ω × [0, T )) = ψ ∈ C∞c (Ω × R,R+), ψ(·, T ) = 0. For a givenm, let us denoteD = Dm, and let(un+1

K )K∈M,n∈[[0,N ]] be the solution of the scheme (3.5)-(3.10) associated toD. Let Ψ = (ΨnK)K∈M,n∈[[0,N+1]]

be defined by

ΨnK = ψ(xK , tn) ∀K ∈M, ∀n ∈ [[0, N + 1]]. (3.60)

Remark 3.5 One cannot use forΨnK the mean value ofψ onK × (tn, tn+1); indeed, in order to pass to the limit

on the termTD13 below (see (3.65)and (3.66)), we shall use the consistency of the approximationΨnK−Ψn

L

d(xK ,xL) to thenormal derivative∇ψ · nK,L. This consistency holds ifΨn

K = ψ(xK , tn) thanks to the assumption on the family(xK)K∈M in Definition 1.4, but does not generally hold ifΨn

K is the mean value ofψ on K × (tn, tn+1). Notethat discrete values using the mean values were used foru when studying an upper bound ofND(u) with respectto theL2(0, T ; H1(Ω)) norm ofu. However we did not have to use the consistency of the flux onu.

With the notations of lemmas 3.2 and 3.3, let us multiply the discrete entropy inequalities (3.17) and (3.19) byδtnΨn

K and sum overK ∈M andn ∈ [[0, N ]]. From (3.17), one gets

TD11 + TD12 + TD13 + TD14 ≤ 0 (3.61)

whereTD11, TD12, TD13 andTD14 are defined by

TD11 =N∑

n=0

δtn∑

K∈Mm(K)

µ(un+1K )− µ(un

K)δtn

ΨnK

TD12 = −N∑

n=0

δtn∑

K∈M

∑

σ∈EK

(qn+1K,σ )−δn+1

K,σ ν(u) ΨnK

TD13 = −N∑

n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dσ

δn+1K,σ η(ϕ(u)) Ψn

K

TD14 =N∑

n=0

δtn∑

K∈M

12

∑

σ∈EK

m(σ)dσ

η′′(ϕ(un+1K,σ ))(δn+1

K,σ ϕ(u))2ΨnK

50

Each of these terms will be shown to converge to the corresponding continuous terms of Inequality (3.59) bypassing to the limit on the space and time steps, i.e. lettingm →∞.

Sinceψ(·, T ) = 0, one hasΨN+1K = 0 and therefore:

TD11 =N∑

n=0

δtn∑

K∈Mm(K)µ(un+1

K )Ψn

K −Ψn+1K

δtn

−∑

K∈Mm(K)Ψ0

Kµ(u0K)

The sequenceµ(uD) converges weakly to∫ 1

0µ(u(·, α))dα asm → ∞. Let χD be the function defined almost

everywhere onΩ × (0, T ) by χD(x, t) = ΨnK−Ψn+1

K

δtn if (x, t) ∈ K × (tn, tn+1); thenχD converges toψt inL1(Ω× (0, T )) asm → +∞. Furthermore, letψ0

M (respu0M) be defined almost everywhere onΩ by ψ0

M = Ψ0K

(resp.u0M = u0

K) if x ∈ K. Then,µ(u0M) converges toµ(u0) in Lp(Ω) for anyp ∈ [1, +∞) andψ0

M convergesto ψ(., 0) uniformly asm → +∞. Hence passing to the limit asm → +∞ in TD11 yields:

limm→∞

TDm11 = −

∫ T

0

∫

Ω

∫ 1

0

µ(u(x, t, α))dα ψt(x, t)dxdt−∫ T

0

∫

Ω

µ(u0(x))ψ(x, 0)dx. (3.62)

Let us now rewriteTD12 as:

TD12 = −N∑

n=0

δtn∑

K∈Mν(un+1

K )∑

σ∈EK

((qn+1K,σ )+Ψn

K,σ − (qn+1K,σ )−Ψn

K). (3.63)

We replace the term(qn+1K,σ )+Ψn

K,σ− (qn+1K,σ )−Ψn

K by 1δtn

∫ tn+1

tn

∫σ

ψ(x, t)q(x, t) ·nK,σdγ(x)dt. When doing so,we commit an error which may be controlled (see the details in [17]) thanks to the consistency and the conserva-tivity of the scheme and thanks to the the weak BV inequality (3.32). Using the weak convergence ofν(uM) to∫ 1

0ν(u(·, α))dα asm →∞, we then obtain:

limm→∞

TDm12 = −

∫ T

0

∫

Ω

∫ 1

0

ν(u(x, t, α))dα∇(q(x, t)ψ(x, t))dxdt

= −∫ T

0

∫

Ω

∫ 1

0

ν(u(x, t, α))dαq(x, t) · ∇ψ(x, t)dxdt. (3.64)

Turning now to the study ofTD13, one remarks that forh(M) small enough, the support ofψ does not intersect thecontrol volumes with edges on∂Ω. Then for all control volumesK ∈ M the sum overσ ∈ EK,ext vanishes andthus

TD13 = −N∑

n=0

δtn∑

K∈M

∑

σ∈EK

m(σ)dK,σδK,ση(ϕ(u))

dσ

ΨnK,σ −Ψn

K

dσ(3.65)

Using the straightforward extension of Lemma 5.13 to time-dependent problems, one gets with computationssimilar as in [19]:

limm→∞

TDm13 = −

∫ T

0

∫

Ω

η(ϕ(u))(x, t)∆ψ(x, t)dxdt =∫ T

0

∫

Ω

∇η(ϕ(u))(x, t) · ∇ψ(x, t)dxdt. (3.66)

51

One now deals withTD14. We remark that

12

∑

K∈M

∑

σ∈EK

m(σ)dσ

η′′(ϕ(un+1K,σ ))(δn+1

K,σ ϕ(u))2ΨnK

=12

∑

K∈M

∑

σ∈EK

m(σ)dση′′(ϕ(un+1K,σ ))

(δn+1K,σ ϕ(u)

dσ

)2

ΨnK

=∑

K∈M

∑

σ∈EK

m(σ)dK,ση′′(ϕ(un+1K,σ ))

(δn+1K,σ ϕ(u)

dσ

)2Ψn

K + ΨnK,σ

2

Thanks to the strong convergence of the function defined inLD(Ω)× (0, T ) by the valueη′′(ϕ(un+1K,σ )) Ψn

K+ΨnK,σ

2to η′′(ϕ(u))ψ, the application of Lemma 5.13 provides

lim infm→∞

TDm14 ≥

∫ T

0

∫

Ω

(∇ϕ(u)(x, t))2η′′(ϕ(u)(x, t))ψ(x, t)dxdt. (3.67)

Gathering (3.61), (3.62), (3.64), (3.66) and (3.67), the proof thatu verifies (3.59) is therefore complete.

The same steps are completed in a similar way in order to show thatu satisfies (3.60), without the difficult problemof the treatment ofη′′. This also completes the proof of Theorem 3.3.¤

To complete the proof of Theorem 3.1 there only remains to show the uniqueness of an entropy process solution.This is the aim of Section 3.2.5.

3.2.5 Uniqueness of the entropy process solution.

One proves in this section the following theorem.

Theorem 3.4 (Uniqueness of the entropy process solution)Under Assumption 3.1, letu and v be two entropyprocess solutions to (3.1)-(3.3) in the sense of Definition 3.3. Then there exists a unique functionw ∈ L∞(Ω ×(0, T )) such thatu(x, t, α) = v(x, t, β) = w(x, t), for almost every(x, t, α, β) ∈ Ω× (0, T )× (0, 1)× (0, 1).

Proof.This proof uses on the one hand Carrillo’s handling of Krushkov entropies, on the other hand the concept ofentropy process solution, which allows the use of the theorem of continuity in means, necessary to pass to the limiton mollifiers. Note that the hypothesis (3.4) makes it easier to handle the boundary conditions.

In order to prove Theorem 3.4, one defines for allε > 0 a regularizationSε ∈ C1(R,R) of the functionsign givenby

Sε(a) = −1, ∀a ∈ (−∞,−ε],Sε(a) = 3ε2a−a3

2 ε3 , ∀a ∈ [−ε, ε],Sε(a) = 1, ∀a ∈ [ε, +∞).

(3.68)

One definesRϕ = a ∈ R,∀b ∈ R \ a, ϕ(b) 6= ϕ(a). Note thatϕ(R \ Rϕ) is countable, because for alls ∈ ϕ(R \ Rϕ), there exists(a, b) ∈ R2 with a < b andϕ((a, b)) = s, and therefore there exists at least oner ∈ Q with r ∈ (a, b) verifying ϕ(r) = s.

52

Let κ ∈ Rϕ. Let ε > 0 and letu be an entropy processus solution. One introduces in (3.59) the functionηε,κ(a) =

∫ a

ϕ(κ)Sε(s − ϕ(κ))ds. One definesµε,κ(a) =

∫ a

κη′ε,κ(ϕ(s))ds andνε,κ(a) =

∫ a

κη′ε,κ(ϕ(s))f ′(s)ds,

for all a ∈ R. Using the dominated convergence theorem, one gets for alla ∈ R that limε→0

ηε,κ(a) = |a− ϕ(κ)|,and, sinceκ ∈ Rϕ, lim

ε→0µε,κ(a) = |a − κ| and lim

ε→0νε,κ(a) = f(a>κ) − f(a⊥κ). One gets for allψ ∈ D+(Ω ×

[0, T )),

∫

Ω×(0,T )

∫ 1

0|u(x, t, α)− κ|dα ψt(x, t)

+∫ 1

0(f(u(x, t, α)>κ)− f(u(x, t, α)⊥κ))dα q(x, t) · ∇ψ(x, t)

−Sε(ϕ(u)(x, t)− ϕ(κ))∇ϕ(u)(x, t) · ∇ψ(x, t)

dxdt

−∫

Ω×(0,T )

[S′ε(ϕ(u)(x, t)− ϕ(κ))(∇ϕ(u))2(x, t)ψ(x, t)

]dxdt

+∫

Ω

|u0(x)− κ|ψ(x, 0)dx ≥ A(ε, u, κ, ψ),

(3.69)where for any entropy process solutionu, anyψ ∈ D+(Ω × [0, T )), anyκ ∈ Rϕ and anyε > 0, A(ε, u, κ, ψ) isdefined by

A(ε, u, κ, ψ) =∫

Ω×(0,T )

∫ 1

0

(|u(x, t, α)− κ| − µε,κ(u(x, t, α))

)dα ψt(x, t)+

∫ 1

0

((f(u(x, t, α)>κ)− f(u(x, t, α)⊥κ))− νε,κ(u(x, t, α))

)dα

q(x, t) · ∇ψ(x, t)

dxdt

+∫

Ω

(|u0(x)− κ| − µε,κ(u0(x))

)ψ(x, 0)dx.

(3.70)Thanks to the dominated convergence theorem, one has

limε→0

A(ε, u, κ, ψ) = 0. (3.71)

This convergence is not uniform w.r.t.κ (even ifκ remains bounded), butA(ε, u, κ, ψ) remains bounded (for agivenu) if κ, ψ, ψt and∇ψ remain bounded and if the support ofψ remains in a fixed compact set ofRd× [0, T ).Using (3.60), one now remarks that, for allκ ∈ R, one has for allψ ∈ D+(Ω× [0, T )),

∫

Ω×(0,T )

∫ 1

0|u(x, t, α)− κ|dα ψt(x, t)+∫ 1

0(f(u(x, t, α)>κ)− f(u(x, t, α)⊥κ))dα

q(x, t) · ∇ψ(x, t)−Sε(ϕ(u)(x, t)− ϕ(κ))∇ϕ(u)(x, t) · ∇ψ(x, t)

dxdt

+∫Ω|u0(x)− κ|ψ(x, 0)dx ≥ B(ε, u, κ, ψ),

(3.72)

where for an entropy process solutionu, all ψ ∈ D+(Ω× [0, T )), all κ ∈ R and allε > 0, B(ε, u, κ, ψ) is definedby

B(ε, u, κ, ψ) =∫

Ω×(0,T )

[∇

(|ϕ(u)(x, t)− ϕ(κ)| − ηε,κ(ϕ(u)(x, t))

)· ∇ψ(x, t)

]dxdt. (3.73)

For allψ ∈ D+(Ω× [0, T )), one has

B(ε, u, κ, ψ) = −∫

Ω×(0,T )

[(|ϕ(u)(x, t)− ϕ(κ)| − ηε,κ(ϕ(u)(x, t))

)∆ψ(x, t)

]dxdt, (3.74)

and

53

limε→0

B(ε, u, κ, ψ) = 0, (3.75)

for all ψ ∈ D+(Ω× [0, T )), ε > 0 andκ ∈ R.As for the study ofA, the quantityB(ε, u, κ, ψ) remains bounded (for a givenu) if κ and∆ψ remain bounded andif the support ofψ remains in a fixed compact set ofRd × [0, T ).

Let u andv be two entropy process solutions in the sense of Definition 3.3. One defines the setsEu = (x, t) ∈Ω × (0, T ), u(x, t, α) ∈ Rϕ, for a.e. α ∈ (0, 1) and Ev = (x, t) ∈ Ω × (0, T ), v(x, t, α) ∈ Rϕ, fora.e. α ∈ (0, 1). Indeed, recall thatϕ(u) andϕ(v) do not depend ofα ∈ (0, 1). Then,Ω × (0, T ) \ Eu =∪s∈ϕ(R\Rϕ)Es,u with Es,u = (x, t) ∈ Ω × (0, T ), ϕ(u)(x, t) = s (the same property is available forv). Letξ ∈ C∞c (Rd × R × Rd × R,R+) such that, for all(x, t) ∈ Ω × [0, T ), ξ(x, t, ·, ·) ∈ D+(Ω × [0, T )) and for all(y, s) ∈ Ω× [0, T ), ξ(·, ·, y, s) ∈ D+(Ω× [0, T )). One introduces in (3.69), for(y, s) ∈ Ev, and a.e.β ∈ (0, 1),κ = v(y, s, β) andψ = ξ(·, ·, y, s). One integrates the result onEv × (0, 1). One then gets

∫

Ev

∫

Ω×(0,T )

∫ 1

0

∫ 1

0|u(x, t, α)− v(y, s, β)|dαdβ ξt(x, t, y, s)+∫ 1

0

∫ 1

0(f(u(x, t, α)>v(y, s, β))− f(u(x, t, α)⊥v(y, s, β)))dαdβ

q(x, t) · ∇xξ(x, t,y, s)−Sε(ϕ(u)(x, t)− ϕ(v)(y, s))∇ϕ(u)(x, t) · ∇xξ(x, t, y, s)

dxdtdyds

−∫

Ev

∫

Ω×(0,T )

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))(∇ϕ(u))2(x, t)ξ(x, t,y, s)

]dxdtdyds

+∫

Ev

∫

Ω

∫ 1

0

|u0(x)− v(y, s, β)|ξ(x, 0, y, s)dβdxdyds

≥∫ 1

0

∫

Ev

A(ε, u, v(y, s, β), ξ(·, ·, y, s))dydsdβ.

(3.76)

One introduces in (3.72), for(y, s) ∈ Ω × (0, T ) \ Ev, and anyβ ∈ (0, 1), κ = v(y, s, β) andψ = ξ(·, ·, y, s).One integrates the result on(Ω× (0, T ) \ Ev)× (0, 1). One then gets

∫

Ω×(0,T )\Ev

∫

Ω×(0,T )

∫ 1

0

∫ 1

0|u(x, t, α)− v(y, s, β)|dαdβ ξt(x, t, y, s)

+∫ 1

0

∫ 1

0(f(u(x, t, α)>v(y, s, β))− f(u(x, t, α)⊥v(y, s, β)))dαdβ

q(x, t) · ∇xξ(x, t, y, s)−Sε(ϕ(u)(x, t)− ϕ(v)(y, s))∇ϕ(u)(x, t) · ∇xξ(x, t,y, s)

dxdtdyds

+∫

Ω×(0,T )\Ev

∫

Ω

∫ 1

0

|u0(x)− v(y, s, β)|ξ(x, 0, y, s)dβdxdyds

≥∫ 1

0

∫

Ω×(0,T )\Ev

B(ε, u, v(y, s, β), ξ(·, ·,y, s))dydsdβ.

(3.77)Adding (3.76) and (3.77) gives

54

∫

Ω×(0,T )

∫

Ω×(0,T )

∫ 1

0

∫ 1

0|u(x, t, α)− v(y, s, β)|dαdβ ξt(x, t,y, s)

+∫ 1

0

∫ 1

0(f(u(x, t, α)>v(y, s, β))− f(u(x, t, α)⊥v(y, s, β)))dαdβ

q(x, t) · ∇xξ(x, t, y, s)−Sε(ϕ(u)(x, t)− ϕ(v)(y, s))∇ϕ(u)(x, t) · ∇xξ(x, t, y, s)

dxdtdyds

−∫

Ev

∫

Ω×(0,T )

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))(∇ϕ(u))2(x, t)ξ(x, t, y, s)

]dxdtdyds

+∫

Ω×(0,T )

∫

Ω

∫ 1

0

|u0(x)− v(y, s, β)|ξ(x, 0,y, s)dβdxdyds

≥∫ 1

0

∫

Ev

A(ε, u, v(y, s, β), ξ(·, ·, y, s))dydsdβ +∫ 1

0

∫

Ω×(0,T )\Ev

B(ε, u, v(y, s, β), ξ(·, ·,y, s))dydsdβ

(3.78)One now exchanges the roles ofu andv, and add the resulting equations. It gives

T15 + T16 + T17(ε) + T18(ε) + T19(ε) ≥ T20(ε), (3.79)

whereT15, T16, T17, T18, T19 andT20 are defined by

T15 =∫

Ω×(0,T )

∫

Ω×(0,T )

∫ 1

0

∫ 1

0|u(x, t, α)− v(y, s, β)|dαdβ (ξt(x, t, y, s) + ξs(x, t, y, s))

+∫ 1

0

∫ 1

0(f(u(x, t, α)>v(y, s, β))− f(u(x, t, α)⊥v(y, s, β)))dαdβ(

q(x, t) · ∇xξ(x, t, y, s) + q(y, s) · ∇yξ(x, t, y, s))

dxdtdyds,

(3.80)

T16 =∫

Ω×(0,T )

∫

Ω

∫ 1

0

|u0(x)− v(y, s, β)|ξ(x, 0, y, s)dβdxdyds

+∫

Ω×(0,T )

∫

Ω

∫ 1

0

|u0(y)− u(x, t, α)|ξ(x, t,y, 0)dαdydxdt,

(3.81)

T3(ε) =

−∫

Ω×(0,T )

∫

Ω×(0,T )

[Sε(ϕ(u)(x, t)− ϕ(v)(y, s))∇ϕ(u)(x, t)·(∇xξ(x, t,y, s) +∇yξ(x, t, y, s))

]dxdtdyds

−∫

Ω×(0,T )

∫

Ω×(0,T )

[Sε(ϕ(v)(y, s)− ϕ(u)(x, t))∇ϕ(v)(y, s)·(∇xξ(x, t,y, s) +∇yξ(x, t, y, s))

]dxdtdyds,

(3.82)

T4(ε) =∫

Ω×(0,T )

∫

Ω×(0,T )

[Sε(ϕ(u)(x, t)− ϕ(v)(y, s))∇ϕ(u)(x, t) · ∇yξ(x, t, y, s)] dxdtdyds

+∫

Ω×(0,T )

∫

Ω×(0,T )

[Sε(ϕ(v)(y, s)− ϕ(u)(x, t))∇ϕ(v)(y, s) · ∇xξ(x, t, y, s)] dxdtdyds,

(3.83)

T19(ε) =

−∫

Ev

∫

Ω×(0,T )

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))(∇ϕ(u))2(x, t)ξ(x, t, y, s)

]dxdtdyds

−∫

Ω×(0,T )

∫

Eu

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))(∇ϕ(v))2(y, s)ξ(x, t, y, s)

]dxdtdyds,

(3.84)

55

and

T20(ε) =∫ 1

0

∫

Ev

A(ε, u, v(y, s, β), ξ(·, ·,y, s))dydsdβ

+∫ 1

0

∫

Ω×(0,T )\Ev

B(ε, u, v(y, s, β), ξ(·, ·, y, s))dydsdβ

+∫ 1

0

∫

Eu

A(ε, v, u(x, t, α), ξ(x, t, ·, ·))dxdtdα

+∫ 1

0

∫

Ω×(0,T )\Eu

B(ε, v, u(x, t, α), ξ(x, t, ·, ·))dxdtdα.

(3.85)

By an integration by parts in (3.83) and using the fact thatξ vanishes on∂Ω × (0, T ) × Ω × (0, T ) and onΩ× (0, T )× ∂Ω× (0, T ) one gets

T4(ε) =∫

Ω×(0,T )

∫

Ω×(0,T )

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))ξ(x, t,y, s)∇ϕ(u)(x, t) · ∇ϕ(v)(y, s)] dxdtdyds

+∫

Ω×(0,T )

∫

Ω×(0,T )

[S′ε(ϕ(v)(y, s)− ϕ(u)(x, t))ξ(x, t, y, s)∇ϕ(v)(y, s) · ∇ϕ(u)(x, t)] dxdtdyds.

(3.86)

Recall thatEs,u = (x, t) ∈ Ω× (0, T ), ϕ(u)(x, t) = s for all s ∈ R. One has∇ϕ(u) = 0 a.e. onEs,u (see [5]for instance). SinceΩ×(0, T )\Eu = ∪s∈ϕ(R\Rϕ)Es,u, and sinceϕ(R\Rϕ) is countable, the following equationshold.

∇ϕ(u) = 0, a.e. onΩ× (0, T ) \ Eu (3.87)

and

∇ϕ(v) = 0, a.e. onΩ× (0, T ) \ Ev. (3.88)

It leads to

T4(ε) =∫

Eu×Ev

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))ξ(x, t, y, s)∇ϕ(u)(x, t) · ∇ϕ(v)(y, s)] dxdtdyds

+∫

Eu×Ev

[S′ε(ϕ(v)(y, s)− ϕ(u)(x, t))ξ(x, t, y, s)∇ϕ(v)(y, s) · ∇ϕ(u)(x, t)] dxdtdyds(3.89)

and

T19(ε) = −∫

Eu×Ev

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))(∇ϕ(u))2(x, t)ξ(x, t, y, s)

]dxdtdyds

−∫

Eu×Ev

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))(∇ϕ(v))2(y, s)ξ(x, t, y, s)

]dxdtdyds.

(3.90)

Therefore∀ε > 0,

T18(ε) + T5(ε)

= −∫

Ev

∫

Eu

[S′ε(ϕ(u)(x, t)− ϕ(v)(y, s))ξ(x, t, y, s)

(∇ϕ(u)(x, t)−∇ϕ(v)(y, s)

)2]

dxdtdyds

≤ 0.

(3.91)

One thus gets∀ε > 0,T15 + T16 + T17(ε) ≥ T20(ε). (3.92)

56

One can now letε → 0 in (3.92). This gives, sinceT20(ε) → 0 (thanks to the dominated convergence theorem),

∫

Ω×(0,T )

∫

Ω×(0,T )

∫ 1

0

∫ 1

0|u(x, t, α)− v(y, s, β)|dαdβ (ξt(x, t, y, s) + ξs(x, t, y, s))+∫ 1

0

∫ 1

0(f(u(x, t, α)>v(y, s, β))− f(u(x, t, α)⊥v(y, s, β)))dαdβ(

q(x, t) · ∇xξ(x, t, y, s) + q(y, s) · ∇yξ(x, t, y, s))

−(∇x|ϕ(u)(x, t)− ϕ(v)(y, s)|+∇y|ϕ(u)(x, t)− ϕ(v)(y, s)|)·(∇xξ(x, t, y, s) +∇yξ(x, t, y, s))

dxdtdyds

+∫

Ω×(0,T )

∫

Ω

∫ 1

0

|u0(x)− v(y, s, β)|ξ(x, 0,y, s)dβdxdyds

+∫

Ω×(0,T )

∫

Ω

∫ 1

0

|u0(y)− u(x, t, α)|ξ(x, t,y, 0)dαdydxdt

≥ 0.(3.93)

Now, let us consider the analog of (3.60) forv instead ofu, with κ = u0(x) andψ(y, s) =∫ T

sξ(x, 0,y, τ)dτ and

integrate the result onx ∈ Ω. One then gets

∫Ω

∫

Ω×(0,T )

− ∫ 1

0|v(y, s, β)− u0(x)|dβ ξ(x, 0,y, s)+∫ 1

0(f(v(y, s, β)>u0(x))− f(v(y, s, β)⊥u0(x)))dβ q(y, s)·

∇y

∫ T

sξ(x, 0, y, τ)dτ

−∇y|ϕ(v)(y, s)− ϕ(u0(x))|·∫ T

s∇yξ(x, 0, y, τ)dτ

dydsdx +

∫

Ω

∫

Ω

|u0(x)− u0(y)|∫ T

0

ξ(x, 0, y, τ)dτdxdy ≥ 0.

(3.94)

A sequence of mollifiers inR andRd is now introduced. Letρ ∈ C∞c (Rd,R+) andρ ∈ C∞c (R,R+) be such that

x ∈ Rd; ρ(x) 6= 0 ⊂ x ∈ Rd; |x| ≤ 1,

x ∈ R; ρ(x) 6= 0 ⊂ [−1, 0] (3.95)

and∫

Rd

ρ(x)dx = 1,

∫

Rρ(x)dx = 1. (3.96)

Forn ∈ N?, defineρn = ndρ(nx) for all x ∈ Rd andρn = nρ(nx) for all x ∈ R.

One setsξ(x, t, y, s) = ψ(x, t)ρn(x − y)ρm(t − s), whereψ ∈ C∞c (Ω × [0, T ),R+) andn andm are largeenough to ensure, for all(x, t) ∈ Ω × [0, T ), ξ(x, t, ·, ·) ∈ D+(Ω × [0, T )) and for all (y, s) ∈ Ω × [0, T ),ξ(·, ·, y, s) ∈ D+(Ω × [0, T )). This choice is not symmetrical in(x, t) and(y, s), which gives an easier way totake the limit asn →∞ andm →∞. One gets, from (3.93),

57

∫

Ω×(0,T )

∫

Ω×(0,T )

ρn(x− y)ρm(t− s)∫ 1

0

∫ 1

0|u(x, t, α)− v(y, s, β)|dαdβ ψt(x, t)

− ∫ 1

0

∫ 1

0

(f(u(x, t, α)>v(y, s, β))−f(u(x, t, α)⊥v(y, s, β))

)dαdβ

(ρn(x− y)ρm(t− s)q(x, t) · ∇ψ(x, t)−ψ(x, t)ρm(t− s)(q(x, t)− q(y, s)) · ∇ρn(x− y))−ρn(x− y)ρm(t− s)(∇x|ϕ(u)(x, t)− ϕ(v)(y, s)|+∇y|ϕ(u)(x, t)− ϕ(v)(y, s)|) · ∇ψ(x, t)

dxdtdyds

+∫

Ω×(0,T )

∫

Ω

∫ 1

0

|u0(x)− v(y, s, β)|ψ(x, 0)ρn(x− y)ρm(−s)dβdxdyds ≥ 0.

(3.97)

The second of the two initial terms vanishes because of the asymmetric choice ofρm. Using the same test functionin (3.94), att = 0, i.e. ξ(x, 0,y, s) = ψ(x, 0)ρn(x− y)ρm(−s) and (3.96), we get

∫

Ω

∫

Ω×(0,T )

− ∫ 1

0|v(y, s, β)− u0(x)|dβ ψ(x, 0)ρn(x− y)ρn(−s)

−∫ 1

0

(f(v(y, s, β)>u0(x))− f(v(y, s, β)⊥u0(x)))dβ q(y, s)·

ψ(x, 0)∇ρn(x− y)∫ T

s

ρm(−τ)dτ

+∇y|ϕ(v)(y, s)− ϕ(u0(x))|·ψ(x, 0)∇ρn(x− y)

∫ T

s

ρm(−τ)dτ

dydsdx

+∫

Ω

∫

Ω

|u0(x)− u0(y)|ψ(x, 0)ρn(x− y)dxdy ≥ 0.

(3.98)

One can now add (3.97) and (3.98) letm tend to∞ and use the theorem of continuity in means. Since the functions → ∫ T

sρm(−τ)dτ is bounded and tends to zero asm →∞ for all s ∈ (0, T ), one gets

∫

Ω

∫

Ω×(0,T )

ρn(y − x)∫ 1

0

∫ 1

0|u(x, t, α)− v(y, t, β)|dαdβ ψt(x, t)

+∫ 1

0

∫ 1

0

(f(u(x, t, α)>v(y, t, β))−f(u(x, t, α)⊥v(y, t, β))

)dαdβ

(ρn(y − x)q(x, t) · ∇ψ(x, t)+ψ(x, t)(q(y, t)− q(x, t)) · ∇ρn(y − x))−ρn(x− y)(∇x|ϕ(u)(x, t)− ϕ(v)(y, t)|+∇y|ϕ(u)(x, t)− ϕ(v)(y, t)|) · ∇ψ(x, t)

dxdtdy

+∫

Ω

∫

Ω

|u0(x)− u0(y)|ψ(x, 0)ρn(x− y)dxdy ≥ 0.

(3.99)

Remarking that∫

Ω

∫

Ω×(0,T )

[ρn(x− y)(∇x|ϕ(u)(x, t)− ϕ(v)(y, t)|+∇y|ϕ(u)(x, t)− ϕ(v)(y, t)|) · ∇ψ(x, t)

]dxdtdy

= −∫

Ω

∫

Ω×(0,T )

[ρn(x− y)|ϕ(u)(x, t)− ϕ(v)(y, t)|∆ψ(x, t)

]dxdtdy,

(3.100)

it is possible to letn →∞ in (3.99). Usingdivq = 0 and the theorem of continuity in means again, one gets

∫

Ω×(0,T )

∫ 1

0

∫ 1

0|u(x, t, α)− v(x, t, β)|dαdβ ψt(x, t)

+∫ 1

0

∫ 1

0(f(u(x, t, α)>v(x, t, β))− f(u(x, t, α)⊥v(x, t, β)))dαdβ

q(x, t) · ∇ψ(x, t)−∇|ϕ(u)(x, t)− ϕ(v)(x, t)| · ∇ψ(x, t)

dxdt ≥ 0. (3.101)

58

One notices that (3.101) holds for anyψ ∈ H1(Ω×(0, T )), with ψ ≥ 0 andψ(., T ) = 0, using a density argument.Therefore one can now take, in (3.101), forψ the functionsψε(x, t) = (T − t)min(d(x,∂Ω)

ε , 1), for ε > 0.Assume momentarily that for allw ∈ H1

0 (Ω) with w ≥ 0,

lim infε→0

∫

Ω

∇w(x) · ∇min(d(x, ∂Ω)

ε, 1)dx ≥ 0 (3.102)

(The proof of (3.102) is given below).

The expressionq(x, t) · ∇min(d(x,∂Ω)ε , 1) verifies

limε→0

q(x, t) · ∇min(d(x, ∂Ω)

ε, 1) = 0, for a.e.(x, t) ∈ Ω× (0, T ),

and under condition (3.4) (and (H5) of Assumption 3.1) remains bounded independently ofε for a.e. (x, t) ∈Ω× (0, T ). Lettingε → 0, (3.101), withψ = ψε, gives

−∫

Ω×(0,T )

[∫ 1

0

∫ 1

0

|u(x, t, α)− v(x, t, β)|dαdβ

]dxdt ≥ 0,

which finally proves thatu = v and thatu is a classical function of space and time (it does not depend onα).

Proof of (3.102)

Let ε > 0. Let (∂Ωi)i=1,...,N be the faces ofΩ, ni their normal vector outward toΩ, and fori = 1, ...N , let Ωi bethe subset ofΩ such that, for allx ∈ Ωi, d(x, ∂Ωi) < ε andd(x, ∂Ωi) < d(x, ∂Ωj) for all j 6= i. One has

∫

∪Ni=1Ωi

∇w(x) · ∇min(d(x, ∂Ω)/ε, 1)dx =N∑

i=1

∫

Ωi

∇w(x) · ni

εdx.

For eachΩi, let Ωi be the largest cylinder generated byni included inΩi. One denotes by∂Ω′i the face ofΩi

parallel to∂Ωi. Let Ωε be defined byΩε = Ω \ ∪Ni=1Ωi. One hasmeas(Ωε) ≤ C(Ω)ε2 and

∫

Ω

∇w(x) · ∇min(d(x, ∂Ω)/ε, 1)dx ≥N∑

i=1

∫

∂Ω′i

w(x)ε

dγ(x)−∫

Ωε

|∇w(x)|ε

dx.

Thanks to the Cauchy-Schwarz inequality, one gets

(∫

Ωε

|∇w(x)|dx)2 ≤ m(Ωε)∫

Ωε

(∇w(x))2dx.

One concludes, usinglimε→0

∫

Ωε

(∇w(x))2dx = 0.

Remark 3.6 Inequation (3.102)could also be proven in the case whereΩ is regular instead of polygonal, with aslightly different method. LetΩε = x ∈ Ω, d(x, ∂Ω) < ε and let∂Ω′ε be the other face ofΩε. The normalvector to∂Ω′ε at any pointx is equal to∇d(x, ∂Ω). Therefore one has

∫

Ω

∇w(x) · ∇min(d(x, ∂Ω)/ε, 1)dx =∫

∂Ω′ε

w(x)ε

dγ(x)−∫

Ωε

w(x)∆d(x, ∂Ω)

εdx.

Since Hardy’s inequality leads to∫

Ωε

(w(x)

d(x, ∂Ω)

)2

dx ≤ C(Ω)∫

Ωε

(∇w(x))2dx,

one concludes usinglimε→0

∫

Ωε

(∇w(x))2dx = 0.

59

¤

3.2.6 Conclusion

Let us finally prove the convergence theorem by way of contradiction:Assume that the convergence stated in the Theorem 3.1 does not hold. Then there existε > 0, p ∈ [1, +∞) anda sequence(uDm)m∈N such that‖uDm − u‖Lp(Ω×(0,T ) ≥ ε, for anym ∈ N. Then by Theorem 3.3, there existsa subsequence of the sequence(uDm

)m∈N, still denoted by(uDm)m∈N which converges to an entropy process

solution of (3.1)-(3.3). By Theorem 3.4 this entropy process solution is the unique entropy weak solution to (3.1)-(3.3), and from Lemma 3.7 which is stated below, the convergence of(uDm

)m∈N is strong in anyLq(Ω× (0, T )).This is in contradiction with the fact that‖uDm

− u‖Lp(Ω×(0,T ) ≥ ε, for anym ∈ N.

Lemma 3.7 Let Q be a Borelian subset ofRk, k ∈ N?, and let(un)n∈N ⊂ L∞(Q) be such thatun converges tou ∈ L∞(Ω × (0, T ))(Q × (0, 1)) in the nonlinear weak star sense whereu does not depend onα, then(un)n∈Nconverges tou in Lp

loc(Q) for anyp ∈ [1,∞).

Proof. Let K be a compact subset ofQ, sinceun converges tou in the nonlinear weak star sense , one has∫

K

|un(x)− u(x)|2dx =∫

K

u2n(x)dx− 2

∫

K

un(x)u(x)dx +∫

K

u(x)2dx → 0 asn → +∞;

sinceK is bounded, one also has:∫

K

|un(x)− u(x)|pdx → 0 asn → +∞, ∀p ∈ [1, 2]

and since the sequence(un)n∈N is bounded inL∞(Q),∫

K

|un(x)− u(x)|pdx → 0 asn → +∞, ∀p > 2.

¤

Remark 3.7 An interesting (and open to our knowledge) question is to find the convergence rate of the finitevolume approximations. In the case of a pure hyperbolic equation, i.e.ϕ = 0, it was proven by several authors(under varying assumptions, see e.g. [12], [34], [17], [9]) that the error between the approximate finite volumesolution and the entropy weak solution is of order less thanh1/4 whereh is the size of the mesh, under a usual CFLcondition for the explicit schemes which are considered in [12], [34], [17], [9], and of order less thanh1/4 + k1/2

wherek is the time step in the case of the implicit scheme considered in [17]. However, it is also known that theseestimates are not sharp, since numerically the order of the error behaves as1/2.

In the case of a pure linear parabolic equation, estimates of order 1 were obtained in [28] (see also [20])

We made a first attempt in the direction of an error estimate in the case of the present degenerate parabolic equationby looking at the analogous continuous problem [21]: letuε be the unique solution to

ut(x, t) + div(q f(u)

)(x, t)−∆ϕ(u)(x, t)− ε∆u(x, t) = 0, for (x, t) ∈ Ω× (0, T ), (3.103)

with initial condition (3.2) and boundary condition (3.3) and letu be the unique entropy weak solution solutionof (3.1)-(3.3), then under Assumption 3.1, we are able to prove that‖uε − u‖L1(QT ) ≤ Cε1/5 whereC ∈ R+

depends only on the data. This estimate is however probably not optimal and we have not yet been able totranscribe its proof to the discrete setting (the term−ε∆u being the continuous diffusive representation of thediffusive perturbation introduced by the finite volume scheme).

Chapter 4

Navier-Stokes equations

4.1 Actual problems and schemes

The Stokes problem

We first study the following linear steady problem: find an approximation ofu andp, weak solution to the gener-alized Stokes equations with homogeneous boundary conditions on∂Ω, which read

−ν∆u +∇p = f in Ω,

divu = 0 in Ω,

u = 0 on∂Ω.

(4.1)

For this problem, in addition to (1.5), the following assumptions are made:

ν ∈ (0, +∞), (4.2)

f ∈ L2(Ω)d. (4.3)

We then consider the following weak sense for problem (4.1).

Definition 4.1 (Weak solution to the steady Stokes equations)Under hypotheses ((1.5),(4.2),(4.3)), letE(Ω) bedefined by:

E(Ω) := v = (v(i))i=1,...,d ∈ H10 (Ω)d, divv = 0 a.e. inΩ. (4.4)

Then(u, p) is called a weak solution of (4.1) (see e.g. [33] or [4]) if

u ∈ E(Ω), p ∈ L2(Ω) with∫

Ω

p(x)dx = 0,

ν

∫

Ω

∇u(x) : ∇v(x)dx

−∫

Ω

p(x)divv(x)dx =∫

Ω

f(x) · v(x)dx, ∀v ∈ H10 (Ω)d.

(4.5)

where, for allu,v ∈ H10 (Ω)d and for a.e.x ∈ Ω, we use the following notation:

∇u(x) : ∇v(x) =d∑

i=1

∇u(i)(x) · ∇v(i)(x) .

The existence and uniqueness of the weak solution of (4.1) in the sense of the above definition is a classical result(see e.g. [33] or [4]).

60

61

The steady Navier-Stokes problem

A solution of the incompressible steady Navier-Stokes equations is given by the fieldsu = (u(i))i=1,...,d : Ω → Rd,andp : Ω → R, such that

−ν∆u +∇p + (u · ∇)u = f in Ω,divu = 0 in Ω.

(4.6)

with homogeneous Dirichlet boundary conditions for the velocity, we define the following weak sense.

Definition 4.2 (Weak solution to the steady Navier-Stokes equations)Under hypotheses ((1.5),(4.2),(4.3)), letE(Ω) be defined by (4.4). Then(u, p) is called a weak solution of (4.6) if

u ∈ E(Ω), p ∈ L2(Ω) with∫

Ω

p(x)dx = 0,

ν

∫

Ω

∇u(x) : ∇v(x)dx

−∫

Ω

p(x)divv(x)dx + b(u,u,v) =∫

Ω

f(x) · v(x)dx ∀v ∈ H10 (Ω)d,

(4.7)

where the trilinear formb(., ., .) is defined, for allu,v,w ∈ (H10 (Ω))d, by

b(u, v, w) =∫

Ω

(u(x) · ∇)v(x) ·w(x)dx =d∑

k=1

d∑

i=1

∫

Ω

u(i)(x)∂iv(k)(x)w(k)(x) dx.

Our concern is to propose some finite volume schemes, in addition to the proof of their convergence. The notion offinite volume scheme will be linked with the mass conservation equation, the gradient of pressure and the nonlinearterms.

4.2 Finite volume scheme with staggered variables

The purpose of this section is to propose a family of finite volume schemes, whose unknowns are the pressures inthe control volumes of a mesh, called the ”pressure mesh”, and the normal velocities to the edges of this mesh.LetD be a finite volume discretization ofΩ in the sense of Definition 1.3, devoted to play the role of the ”pressuremesh”. In this section, we consider that, for allσ ∈ E , the point xσ is the center of gravity of σ. We definethe setXE of all u ∈ RE such thatuσ = 0 for all σ ∈ Eext, which contains the discrete velocities, and we definethe setPE = xσ, σ ∈ E. The scheme first consists in findingu ∈ XE andp ∈ HM(Ω) such that the massconservation is discretized by the following finite volume scheme:

divKu = 0, ∀K ∈M, (4.8)

where

divKv =1

m(K)

∑

σ∈EK

m(σ)vK,σ, ∀K ∈M, (4.9)

and where we denotevK,σ = vσnσ · nK,σ, ∀σ ∈ EK , ∀K ∈M, (4.10)

We now discretize the momentum equation, using a variational formulation. We define the setLD,0(Ω) ⊂ LD(Ω)of the piecewise constant functions inDK,σ, such that, for allu ∈ LD,0(Ω), thenuK,σ = 0 for all K ∈ M andσ ∈ EK ∩ Eext. We then assume that the following hypotheses, denoted below by Hypotheses (HS), are fulfilled:

62

1. There exists a linear mappingΠD : XE → LD(Ω) (see definition 1.3), such that, forϕ ∈ C2(Ω)d ∩H1

0 (Ω)d, definingv ∈ Xσ by vσ = ϕ(xσ) · nσ, thenΠK,σv, the constant value ofΠDv in DK,σ, is asecond order approximation at the pointxσ of ϕ(xσ). This is completed, assuming the relation

ΠK,σv =∑

σ′∈Eaσ′

σ vσ′ , ∀v ∈ XE , ∀K ∈M, ∀σ ∈ EK , (4.11)

where the coefficientsaσ′σ are assumed to satisfy

∑

σ′∈Eaσ′

σ (v + A(xσ′ − xσ)) · nσ′ = v, ∀v ∈ Rd, ∀A ∈ Rd×d, ∀K ∈M, ∀σ ∈ EK . (4.12)

We then denoteΠDv = (Π(i)D v)i=1,...,d.

2. The following consistency property holds:

ΠK,σv · nσ = vσ, ∀K ∈M, ∀σ ∈ EK , ∀v ∈ XE . (4.13)

3. An inner product〈·, ·〉D is defined onLD(Ω)d, such that, under suitable hypotheses, it approximates theinner product

∫Ω∇u : ∇vdx.

Then we approximate the equation (4.7) by

ν〈ΠDu,ΠDv〉 −∫

Ω

p(x)divDv(x)dx + bD(u, u, v) =∫

Ω

f(x) · Πv(x)dx, ∀v ∈ XE , (4.14)

with Πv ∈ HM(Ω)d defined by

ΠvK =1

m(K)

∑

σ∈EK

m(σ)vK,σ(xσ − xK), ∀K ∈M, (4.15)

bD(u, v, w) =∑

K∈M

∑σ∈EK

Mσ=K,L

m(σ)uK,σΠvL − ΠvK

2· ΠwK , ∀u, v, w ∈ XE . (4.16)

Hence the number of unknowns is#Eint + #M. Note that the following property holds

bD(u, v, v) = −12

∑

K∈Mm(K)|ΠvK |2divKu, ∀u, v ∈ XE , (4.17)

which mimicks the continuous propertyb(u, v, v) = − 12

∫Ω|v|2divudx.

4.2.1 The MAC scheme on regular grids

Let us briefly show why the MAC scheme can be analyzed following the line of the schemes presented here. Weconsider a grid, whose control volumes are orthogonal parallelepipedic, without grid refinement. In such a case, itis easy to define the inner product〈·, ·〉D as the sum of the contributions of each directioni = 1, . . . , d of scalarproducts, such as the one defined by (2.18), available in the case of∆−adapted pointed meshes.

63

4.2.2 An example, built on the pressure grid

We aim in this section to use the scheme described in section 2.1. We use the matrixAσσ′K defined by (2.45), for

setting, fori = 1, . . . , d,

〈u,v〉D =∑

K∈M

∑

σ∈EK

∑

σ′∈EK

Aσσ′K (uσ − uK) · (vσ′ − vK), ∀u,v ∈ LD,0(Ω)d, (4.18)

with

uK =

∑σ∈EK

uσ

∑σ′∈EK

Aσσ′K∑

(σ,σ′)∈E2K

Aσσ′K

, ∀K ∈M. (4.19)

We then get〈u, v〉D =

∑

K∈M

∑

σ∈EK

∑

σ′∈EK

Aσσ′K uσ · vσ′ , ∀u, v ∈ LD,0(Ω)d, (4.20)

withAσσ′

K = Aσσ′K − aK,σaK,σ′SK (4.21)

where we defineSK =∑

(σ,σ′)∈E2K

Aσσ′K and we set

aK,σ =1

SK

∑

σ′∈EK

Aσσ′K . (4.22)

We define coefficientsα(i)σσ′ such that the mappingΠi is defined by

Πiv = vσ · e(i) =∑

σ′∈Eα

(i)σσ′vσ′ , ∀σ ∈ Eint (4.23)

with the propertiesd∑

i=1

α(i)σσe(i) · nσ = 1 (4.24)

andd∑

i=1

∑

σ′∈Eα

(i)σσ′e

(i) · nσ = 0, ∀σ′ 6= σ, (4.25)

and second order approximation, that is: if there existsx and a matrixB with vσ′ = (x + B(xσ′ − xσ)) ·nσ′ forall σ′ ∈ E , then

∑σ′∈E α

(i)σσ′vσ′ = x · e(i).

Numerical example

We show in the following figure the stream lines, obtained withRe = 5000 on a80× 80 grid.

4.2.3 Further analysis

Let us provide the mathematical analysis of the convergence of the above method. To this purpose, we introducethe following definition for the regularity of the mesh:

θD = max

(max

σ∈Eint,K,L∈Mσ

dK,σ

dL,σ, maxK∈M,σ∈EK

(hK

dK,σ,

∑σ′∈E |aσ′

σ ||xσ′ − xσ|2h2

K

)). (4.26)

Then we define a norm, suitable for deriving compactness properties:We then introduce the notion of continuous, coercive, consistent and symmetric families of inner products onLD(Ω).

64

Figure 4.1: Streamlines for the lid driven cavity withRe = 5000

Definition 4.3 (Continuous, coercive, consistent and symmetric families of inner products)LetF be a familyof discretizations in the sense of definition 1.2. ForD = (M, E ,P) ∈ F , K ∈ M and σ ∈ E , we denote by〈u,v〉D an inner product onLD(Ω)d. The family of inner products(〈·, ·〉D)D∈F is said to be continuous if thereexistsM > 0 such that

(〈u, u〉D)1/2 ≤ M

hD|u|L2(Ω)d , ∀u ∈ LD(Ω)d, ∀D = (M, E ,P) ∈ F . (4.27)

The family of inner products(〈·, ·〉D)D∈F is said to be coercive if for any sequence(Dn)n∈N of elements ofFsatisfying thatlimn→∞ hDn = 0 and that there existsun ∈ LDn(Ω)d for all n ∈ N and there existsC > 0 with〈un,un〉Dn ≤ C for all n ∈ N, then there existsu ∈ H1

0 (Ω)d and a subsequence family of(Dn)n∈N, againdenoted(Dn)n∈N, with lim

hD→0‖uD − u‖L2(Ω) = 0

The family of inner products(〈·, ·〉D)D∈F is said to be consistent (with the viscous operator) if for any family(uD)D∈F satisfying:

• uD ∈ LD(Ω)d for all D ∈ F ,

• there existsC > 0 with 〈uD, uD〉D ≤ C for all D ∈ F ,

• there existsu ∈ H10 (Ω)d with lim

hD→0‖uD − u‖L2(Ω) = 0,

then, for allϕ ∈ C∞c (Ω)d, the following holds

limhD→0

〈uD,vD〉D =∫

Ω

∇u(x) : ∇ϕ(x)dx, (4.28)

where, omitting for the simplicity of notation the indexn, we definevD ∈ LD(Ω)d defined byvσ = 1m(σ)

∫σ

ϕ(x)dγ(x)for all σ ∈ E . Finally the family of inner products(〈·, ·〉D)D∈F is said to be symmetric if

〈u,v〉D = 〈v, u〉D, ∀u,v ∈ LD(Ω)d, ∀D = (M, E ,P) ∈ F .

We then have the following lemma.

Lemma 4.1 (Discrete Poincare inequality) Under hypotheses (1.5), let F be a family of discretizations in thesense of definition 1.2 and let(〈·, ·〉D)D∈F be a coercive family of inner products, in the sense of Definition 4.3.Then, for all sequence(Dn)n∈N of elements ofF such thatlimn→∞ hDn = 0, there existsC1 > 0 such that

‖v‖2L2(Ω)d ≤ C1〈v, v〉Dn , ∀v ∈ LDn(Ω)d, ∀n ∈ N. (4.29)

Proof. For anyn ∈ N, since the dimension ofLDn(Ω) is finite, one can set

ζn = supv∈LDn (Ω)d

‖v‖2L2(Ω)d

〈v, v〉Dn

.

65

Morover, there existsvn ∈ LDn(Ω)d such that〈vn,vn〉Dn = 1 and‖v‖2L2(Ω)d = ζn. Let us assume that thelemma is wrong. It means that the sequence(ζn)n∈N is not bounded. Then there exists a strictly increasingfunctionψ fromN toN, such that the sequence(ζψ(n))n∈N tends to+∞.Then, using the coercivity of the family of inner products, we get from〈vψ(n),vψ(n)〉Dψ(n) = 1 and fromlimn→∞ hDψ(n) = 0 the existence of a subsequence of(Dψ(n))n∈N, denoted(Dψ(ϕ(n)))n∈N, whereϕ is strictlyincreasing fromN toN, and an elementv ∈ H1

0 (Ω)d, such that

limn→∞

‖vψ(ϕ(n)) − v‖L2(Ω)d = 0.

We then get a contradiction with‖vψ(ϕ(n))‖2L2(Ω)d = ζψ(ϕ(n)), which tends to+∞ asn → +∞. ¤We then have the following lemma.

Lemma 4.2 (Estimate) Under hypotheses (1.5)and (4.2), letD be a finite volume discretization ofΩ in the senseof Definition 1.3. LetζD be defined by

ζD = supv∈LD(Ω)d

‖v‖2L2(Ω)d

〈v, v〉Dn

. (4.30)

Let (uE , pM) ∈ XE ×HM(Ω) be a solution to (4.8)-(4.9)-(4.14). Then we have:

ν(〈ΠDu,ΠDu〉D)1/2 ≤ d θDζD‖f‖L2(Ω)d . (4.31)

whereθD is defined by (4.26).

Proof.Let us takev = u in (4.14). We get

ν〈Πu,Πu〉D =∫

Ω

f(x) · Πu(x)dx,

thanks to (4.8), which, together with (4.17), also impliesb(u, u, u) = 0. Denoting byfK = 1m(K)

∫K

f(x)dx, wehave ∫

Ω

f(x) · Πu(x)dx =∑

K∈MfK

∑

σ∈EK

m(σ)uK,σ(xσ − xK).

Applying the Cauchy-Schwarz inequality, we get

(∫

Ω

f(x) · Πu(x)dx

)2

≤∑

K∈M|fK |2

∑

σ∈EK

m(σ)|xσ − xK |∑

K∈M

∑

σ∈EK

m(σ)u2K,σ|xσ − xK |,

which gives, thanks to the definition of the regularity of the mesh,∣∣∣∣∫

Ω

f(x) · Πu(x)dx

∣∣∣∣ ≤ d θD‖f‖L2(Ω)d‖ΠDv‖L2(Ω)d .

We can then writeν〈ΠDu,ΠDu〉D ≤ d θDζD‖f‖L2(Ω)d(〈ΠDu,ΠDu〉D)1/2,

which implies (4.31).¤We then get the following lemma.

Lemma 4.3 (Existence of a discrete solution)Under hypotheses (1.5) and (4.2), let D be a finite volume dis-cretization ofΩ in the sense of Definition 1.3. Then there exists at least one(uE , pM) ∈ XE ×HM(Ω), solutionto (4.8)-(4.9)-(4.14).

66

Proof. It suffices to remark that one can estimate the pressure, using Necas procedure.¤Let us now state the convergence of the scheme.

Theorem 4.1 Under hypotheses (1.5) and (4.2), let F be a family of discretizations in the sense of definition1.3 and let(〈·, ·〉D)D∈F be a coercive family of inner products, in the sense of Definition 4.3. Let(Dn)n∈Nbe a sequence of elements ofF , such thatlimn→∞ hDn

= 0 and such that there existsθ > 0 with θ ≥ θDn

(defined by (4.26)), for all n ∈ N. For all n ∈ N, let (un, pn) ∈ XEn× HMn

(Ω) be a solution to (4.8)-(4.9)-(4.14) with D = Dn. Then there exists a subsequence of(Dn)n∈N, again denoted(Dn)n∈N, and a solution(u, p) ∈ E(Ω)× L2(Ω) of (4.7)such thatΠDn

un tends tou in L2(Ω)d.

Proof. Let us first remark thatζDndefined by (4.30) is bounded thanks to Lemma 4.1 by some value, denoted

by C1. Thanks to Lemma 4.2, we get that the sequence〈ΠDnun,ΠDnun〉Dn is bounded, which shows, from thecoercivity of the family of inner products(〈·, ·〉D)D∈F , that we can extract from(Dn)n∈N a subsequence, againdenoted(Dn)n∈N, such that there existsu ∈ H1

0 (Ω)d with ΠDnun tends tou in L2(Ω)d. We then see, using (4.8)and the consistency property (4.13), thatdivu = 0 a.e. inΩ, which impliesu ∈ E(Ω). Let ϕ ∈ C∞c (Ω)d, withdivϕ = 0 in Ω be given. Letn ∈ N. Omitting for the simplicity of notation the indexn, we definevD ∈ XE byvσ = 1

m(σ)

∫σ

ϕ(x) ·nσdγ(x) for all σ ∈ E . andvD ∈ LD(Ω)d by vσ = 1m(σ)

∫σ

ϕ(x)dγ(x) for all σ ∈ E . Notethat we have, thanks to the definition ofθD,

‖vD −ΠDvD‖L2(Ω)d ≤ h2DθD

max |∂2ijϕb|

2.

Using the consistency of the family of inner products(〈·, ·〉D)D∈F , we get

limn→∞

〈ΠDnun, vDn〉Dn =∫

Ω

∇u(x) : ∇ϕ(x)dx.

Using the continuity of the family of inner products(〈·, ·〉D)D∈F , we have

limn→∞

〈vD −ΠDvD, vD −ΠDvD〉Dn = 0,

which implies

limn→∞

〈ΠDnu,ΠDnvDn〉Dn =∫

Ω

∇u(x) : ∇ϕ(x)dx.

Since we havedivDvD = 0 in this case, we can write, takingvD as test function in (4.14)

ν〈ΠDu,ΠDvD〉D + bD(u, u, vD) =∫

Ω

f(x) · ΠvD(x)dx.

We get, using the results of chapter (5), thatbDn(un, un, vDn) converges tob(u, u, ϕ), using the convergence in

L2(Ω)d of Πun to u, the weak convergence of the piecewise constant function with valuedbΠ(i)

L u−bΠ(i)K u

2dK,σnK,σ in

DK,σ to ∇u(i) and the convergence inL∞(Ω)d of the piecewise constant function with valuebΠLvD+bΠKvD

2 inDK,σ to ϕ. The proof is concluded thanks to the convergence of the right hand side.¤ We then have the following lemma.

Lemma 4.4 (Application) The scheme defined by (4.18)-(4.22)provides a continuous, coercive, consistent andsymmetric families of inner products.

We let the proof of the lemma to the reader, using the results of Chapter 2.

67

4.3 Finite volume scheme with collocated variables

4.3.1 Discrete scheme

We denote byD = (M, E ,P) a pointed discretization in the sense of Definition 1.2.For all functionψ ∈ C0(Ω), we recall thatPMψ is the elementv ∈ HM(Ω) such thatvK = ψ(xK) for allK ∈ M. Let us follow the scheme presented in Chapter 2, in which we want to directly define the unknowns inHM(Ω) instead ofXD,0. To this purpose, for any edgeσ ∈ Eint, we define a linear mappingΠσ : HM(Ω) → Rsuch that for all regular functionψ, ΠσPMψ is a second order approximation ofψ(xσ). In such a case, followingthe idea of (2.24), we use coefficients(βL

σ )L∈M such that

∀u ∈ HM(Ω), Πσu =∑

L∈MβL

σ uL with∑

L∈MβL

σ = 1, xσ =∑

L∈MβL

σ xL. (4.32)

It is always possible to restrict in 3D (resp. 2D) the number of nonzeroβLσ to four (resp. three). In practice,

the scheme is shown to be robust with respect to the choice of these four control volumes if they are taken closeenough to the considered edge.We introduce some notations related to the mesh. The size of the discretizationD is defined by:

hD = suphK ,K ∈M,

and the regularity of the mesh by:

θD = max

max

σ∈EintK,L∈Mσ

dK,σ

dL,σ, max

K∈Mσ∈EK

hK

dK,σ, max

K∈Mσ∈EK∩Eint

∑

L∈M|βL

σ ||xL − xσ|2

h2K

. (4.33)

Discretization of the viscous terms

We begin with defining an approximate of the gradient of elements ofHM(Ω) on cellK ∈ M. We set, for anyu ∈ HM(Ω) andK ∈M:

∇Ku =1

m(K)

∑

σ∈EK

m(σ)(Πσu− uK)nK,σ. (4.34)

Then, for allσ ∈ EK , we defineRK,σu ∈ R, corresponding to some local residual of the second order interpolationof a regular function, by:

RK,σu =

√d

dK,σ(Πσu− uK −∇Ku · (xσ − xK)) . (4.35)

We then give the following expression for the discrete gradient ofu ∈ HM(Ω) in the coneCK,σ:

∇K,σu = ∇Ku + RK,σunK,σ. (4.36)

We can then define the function∇Du by

∇Du(x) = ∇K,σu, for a.e.x ∈ CK,σ, ∀K ∈M, ∀σ ∈ EK . (4.37)

We then get that∫

Ω

∇Du(x) · ∇Dv(x)dx =∑

K∈M

∑

σ∈EK

m(σ)dK,σ

d∇K,σu · ∇K,σv, ∀u, v ∈ HM(Ω). (4.38)

Notice that∫Ω∇Du(x) · ∇Dv(x)dx defines a symmetric inner product onHM(Ω) which provides a good ap-

proximation for∫Ω∇u(x) · ∇v(x)dx and that the following lemma holds (see lemma 2.5):

68

Lemma 4.5 LetD be a space discretization ofΩ in the sense of Definition 1.2 and letθ ≥ θD. Then, defining

|u|D =

( ∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(Πσu− uK)2)1/2

, ∀u ∈ HM(Ω), (4.39)

there existα > 0 andβ > 0, only depending onθ, such that

α|u|D ≤ ‖∇Du‖L2(Ω)d ≤ β|u|D, ∀u ∈ HM(Ω). (4.40)

Pressure-velocity coupling, mass balance and convective contributions

For all v ∈ (HM(Ω))d, we define a discrete divergence operator by:

divK v =1

m(K)

∑

σ∈EK

m(σ)Πσ v · nK,σ =d∑

i=1

(∇Kv(i))(i), ∀K ∈M. (4.41)

We then define the functiondivDv by the relation

divDv(x) = divK v, for a.e.x ∈ K, ∀K ∈M.

As recalled in the introduction of this paper, a pressure stabilization method must be implemented in the massconservation equation in order to prevent from oscillations of the pressure. To this aim, we first define a stabilizedmass flux acrossσ ∈ Eint with Mσ = K, L, by

ΦλK,σ(u, p) = m(σ) (Πσu · nK,σ + λσ(pK − pL)) , (4.42)

where(λσ)σ∈Eint is a given family of positive reals, the choice of which is discussed in [11], in the case of meshessatisfying an orthogonality property. In this paper we set

λσ = λhγD

dK,σ + dL,σ, ∀σ ∈ Eint with Mσ = K,L, (4.43)

for given valuesλ > 0 andγ ∈ (0, 4) (this expression, chosen in [7] in the framework of a collocated finite elementmethod, makes the mathematical study easier than the ”cluster” choice, which is shown in [11] to be more precise).Note that, for allσ ∈ Eint with Mσ = K, L, Φλ

K,σ(u, p) + ΦλL,σ(u, p) = 0. We then use this modified flux,

in order to define a stabilized transport operator. The centred transport operator is defined, for allv ∈ (HM(Ω))d,w ∈ HM(Ω) andK ∈M, by

divλK(w, v, p) =

1m(K)

∑

σ∈EK∩Eint,Mσ=K,LΦλ

K,σ(u, p)wK + wL

2.

Then the functiondivλD(w, v, p) is defined by

divλD(w, v, p)(x) = divλ

K(w, v, p), for a.e.x ∈ K, ∀K ∈M.

Resulting discrete equations

By considering the previous definitions, the discrete approximation of equations (4.6) therefore reads: findu andp such that

u ∈ (HM(Ω))d andp ∈ HM(Ω) with∫

Ω

p(x)dx =∑

K∈Mm(K)pK = 0,

∫

Ω

(ν∇Du : ∇Dvdx− pdivDv + divλ

D(u,u, p) · v)

dx

=∫

Ω

f · vdx, ∀v ∈ (HM(Ω))d,

(4.44a)

divλD(1, u, p) = 0 a.e. inΩ. (4.44b)

69

4.3.2 Further analysis

Existence of a discrete solution and estimates

The system of discrete equations (4.44) appears as a system of nonlinear equations, for which we have to prove theexistence of at least one solution, satisfying suitable estimates. In this direction, we have the following lemma.

Lemma 4.6 (DiscreteH10 (Ω) estimate on the velocities)LetD be a space discretization ofΩ, let θ ≥ θD and let

λ ∈ (0, +∞) andγ ∈ (0, 4) be given. Letρ ∈ [0, 1] be given and let(u, p), be a solution to the following systemof equations (which reduces to (4.44)asρ = 1)

u ∈ (HM(Ω))d andp ∈ HM(Ω) with∫

Ω

p(x)dx =∑

K∈Mm(K)pK = 0,

∫

Ω

(ν∇Du : ∇Dvdx− pdivDv + ρdivλ

D(u, u, p) · v)

dx

=∫

Ω

f · vdx, ∀v ∈ (HM(Ω))d,

(4.45a)

divλD(1,u, p) = 0 a.e. inΩ. (4.45b)

Then there existsC2, only depending onθ such thatu andp satisfy the following estimates:

ν‖∇Du‖(L2(Ω)d)d ≤ C2‖f‖(L2(Ω))d∑

σ∈Eint,Mσ=K,Lλσm(σ)(pL − pK)2 ≤ C2‖f‖2(L2(Ω))d (4.46)

Proof. We let v = u in (4.45a). Let us first remark that, thanks to (4.45b), we get∫Ω

divλD(u, u, p) · udx = 0

and ∫

Ω

pdivDudx = −∑

σ∈Eint,Mσ=K,Lλσm(σ)(pL − pK)2.

We apply the Cauchy-Schwarz inequality, and the discrete Poincare inequality. We then conclude the proof of thelemma.¤We can now state the existence of at least one solution to (4.44).

Lemma 4.7 (Existence of at least one solution)LetD be a space discretization ofΩ, let λ ∈ (0, +∞) andγ ∈(0, 4) be given. Then there exists at least one(u, p) solution to (4.44), which therefore satisfies (4.46).

Proof. We remark that (4.45) withρ = 0 is a linear system. Thanks to Lemma 4.6, we can then apply the resultson the topological degree [32]. This shows the existence of at least one solution of (4.45) withρ = 1, that is (4.44).¤As in [32], we could also state anL2 estimate on the pressure, under some additionnal regularity hypotheses onthe mesh. For the sake of shortness, we focus here on the convergence of the scheme to the solution of (4.6).

Passing to the limit

Let us now state the convergence theorem.

Theorem 4.2 Let (D(m))m∈N be a sequence of space discretizations ofΩ in the sense of Definition 1.2, such thathD(m) tends to0 asm → ∞ and such that there existsθ > 0 with θD(m) ≤ θ, for all m ∈ N. Letλ ∈ (0, +∞)and γ ∈ (0, 4) be given. Let, for allm ∈ N, (u(m), p(m)) ∈ (XD(m))d × HM(m)(Ω), be a solution to (4.44)with D = D(m). Then there exists a weak solutionu of (4.6) and a subsequence of(D(m))m∈N, again denoted(D(m))m∈N, such that the corresponding subsequence of solutions(u(m))m∈N converges tou in L2(Ω)d.

70

Proof. Thanks to (4.46), we can apply the results of chapter 2 concerning the relative compactness of thefamily (u(m))m∈N and the regularity of the limit of a subsequence. Hence we get the existence of a subsequenceof (D(m))m∈N, such that the corresponding sequence(u(m))m∈N converges to some functionu ∈ H1

0 (Ω)d inL2(Ω)2. Using (4.41), we get thatdivD(m)u(m) converges for the weak topology ofL2(Ω)d to divu. Moreover,the corresponding sequence(p(m))m∈N satisfies for allm ∈ N the inequality (4.46). Hence, multiplying (4.44b)by a regular test function and summing onK ∈ M(m), we get that the term in pressure tends to 0 asm → ∞,using (4.46) and the Cauchy-Schwarz inequality. This proves thatdivu = 0 a.e. inΩ, and therefore thatu ∈ V .We then consider a functionψ ∈ V such thatψ ∈ C∞c (Ω)d and we denote, for somem ∈ N,D = D(m). We thensetv = PMψ in (4.44). We define the terms

T(m)1 =

∫

Ω

ν∇Du : ∇DPMψdx, T(m)2 = −

∫

Ω

pdivDPMψdx,

T(m)3 =

∫

Ω

divλD(u, u, p) · PMψdx, T

(m)4 =

∫

Ω

f · PMψdx.

Using the results of chapter 2, we getlimm→∞

T(m)1 =

∫

Ω

ν∇u : ∇ψdx. We have

∫

Ω

pdivDPMψdx =∑

σ∈Eint,Mσ=K,Lm(σ)(pL − pK)ΠσPMψ · nK,σ.

Usingdivψ = 0, we get, settingψσ = 1m(σ)

∫σ

ψds, that∫

Ω

pdivDPMψdx =∑

σ∈Eint,Mσ=K,Lm(σ)(pL − pK)(ΠσPMψ −ψσ) · nK,σ.

Thanks to the definition of the regularity of the mesh, we have the existence ofC3, only depending onψ andθ,such that|ΠσPMψ−ψσ| ≤ C3h

2D. Therefore, thanks to the Cauchy-Schwarz inequality,γ ∈ (0, 4) and to (4.46),

we getlimm→∞ T(m)2 = 0.

Remark 4.1 In [32], the exponentγ had to be taken lower than2 in the pressure stabilization term, since theconsistency of the divergence operator was only satisfied with an order 1.

We now turn toT (m)3 . Using (4.44b), we rewrite this term asT

(m)3 = T

(m)5 + T

(m)6 with

T(m)5 =

∑

σ∈Eint,Mσ=K,Lm(σ)Πσu(m) · nK,σ(u(m)

L − u(m)K ) · ψ(xK) + ψ(xL)

2

and

T(m)6 =

∑

σ∈Eint,Mσ=K,Lm(σ)λσ(pL − pK)(u(m)

L − u(m)K ) · ψ(xK) + ψ(xL)

2.

We use the fact that, for alli = 1, . . . , d, the function, defined inCK,σ by the constant value ddK,σ

(Πσu(m)L −

u(m)K )(i)nK,σ weakly converges inL2(Ω)2 to ∇u(i) and the function, defined inCK,σ by the valueΠσu(m)

converges inL2(Ω)2 to u, we get

limm→∞

T(m)5 =

∫

Ω

div(u⊗ u) ·ψdx.

Using the Cauchy-Schwarz inequality, we get that there existsC4 > 0, only depending onθ such that

(T (m)6 )2 ≤ hγ

DλC4

∑

σ∈Eint,Mσ=K,Lλσm(σ)(pL − pK)2|u(m)|2D‖ψ‖∞,

which shows, thanks to (4.46) and to (4.40), thatlimm→∞ T(m)6 = 0. We easily get thatlimm→∞ T

(m)4 =

∫Ω

f ·ψdx. Gathering the above results, we get thatu is a solution to (4.6), which concludes the proof.¤

71

4.3.3 Numerical example

We assume that the analytical solution is given by

pref(x) = cos(πx(1)) cos(πx(2)) cos(πx(3))uref(x) = curl(

∑di=1(4x(1)(x(1) − 1))3(4x(2)(x(2) − 1))4(4x(3)(x(3) − 1))5e(i))

in the caseν = 1 and Ω = (0, 1)3. We consider three types of meshes withn3i control volumes, forni =

10, . . . , 60. The first ones are regular cubic, in the second ones (”smooth meshes”), the vertices are slightly movedby a regular nonlinear mapping. In this case, the control volumes become ”hexahedra” with non planar facessplitted into two triangles, which means that the interior control volumes have six neighbors, but that any pairof neighboring control volumes share two faces, and the control volumes are no longer convex. In the ”randommeshes”, the vertices of cubic meshes are randomly moved, only preserving that the faces can be defined withoutexcluding the initial center of the control volumes. Table 4.1 shows that the convergence orders of the gradients

Cubic meshes Smooth meshes Random meshesL2 u 2 1.91 1.71

p 2 1.08 0.80H1 u 1.98 1.77 1.34

p 1.91 −− −−

Table 4.1: Numerical orders of convergence.

are better than the expected first order. Unsurprisingly, theH1-norm of the pressure does not tend to zero with thesize of the mesh.

Chapter 5

Discrete functional analysis

The contents of this chapter concern mathematical analysis tools which are needed for the in depth mathematicalstudies of the following finite volume schemes. It can be bypassed by the reader who is mainly interested in theimplementation features and the numerical efficiency of the finite volume schemes.

5.1 The topological degree argument

This argument is used several times in this book in order to prove the existence of a solution to systems of nonlinearequations provided by finite volume schemes. For the sake of completeness, we recall this argument (which wasfirst used for numerical schemes in [17]) in the finite dimensional case in the following theorem and refer to [14]for the general case.

Theorem 5.1 (Application of the topological degree, finite dimensional case)Let V be a finite dimensional vector space onR andg be a continuous function fromV to V . Let us assume thatthere exists a continuous functionF fromV × [0, 1] to V satisfying:

1. F (·, 1) = g, F (·, 0) is an affine function.

2. There existsR > 0, such that for any(v, ρ) ∈ V × [0, 1], if F (v, ρ) = 0, then‖v‖V 6= R.

3. The equationF (v, 0) = 0 has a solutionv ∈ V such that‖v‖V < R.

Then there exists at least a solutionv ∈ V such thatg(v) = 0 and‖v‖V < R.

5.2 Discrete Sobolev embedding

Note that these properties hold for general finite volume discretizations.

5.2.1 Discrete embedding ofW 1,1(Ω) in L1?(Ω)

Let us first give the discrete counterpart of the norm inW 1,1(Ω).

Definition 5.1 Under hypothesis (1.5), letD be a polygonal finite volume space discretization ofΩ in the sense ofDefinition 1.1. We define a norm onHM(Ω) by

‖u‖1,1,D =∑

σ∈Mm(σ)δσu, ∀u ∈ HM(Ω) (5.1)

definingδσu for all u ∈ HM(Ω) andσ ∈ E by

δσu = |uL − uK |, ∀σ ∈ Eint, Mσ = K, L,δσu = |uK |, ∀σ ∈ Eext, Mσ = K. (5.2)

72

73

We can now state the inequality corresponding to the discrete embedding ofHM(Ω) in L1?

(Ω).

Lemma 5.1 Under hypothesis (1.5), letD be a polygonal finite volume space discretization ofΩ in the sense ofDefinition 1.1. Then, with the notation of Definition 5.1 :

‖u‖L1? (Ω) ≤1

2√

d‖u‖1,1,D, ∀u ∈ HM(Ω), (5.3)

where1? = dd−1 .

Proof.Different proofs of this lemma are possible. A first one consists to adapt to this discrete setting the classical proofof the Sobolev embedding due to L. Nirenberg (actually, it gives1/2 instead of1/(2

√d) in (5.3)). It is done using

an induction ond. This proof is essentially given in [20]. Indeed, the hypotheses given in [20] are slightly lessgeneral but a quite easy adaptation of the proof leads to the present lemma (with1/2 instead of1/(2

√d) in (5.3)).

We give here another proof using directly the result of L. Nirenberg, namely:

‖u‖L1? (Rd) ≤12d‖u‖W 1,1(Rd), ∀u ∈ W 1,1(Rd), (5.4)

where‖u‖W 1,1(Rd) =∑d

i=1 ‖Diu‖L1(Rd) andDiu is the weak derivative (or derivative in the sense of distribu-tions) ofu in the directionxi (with x = (x1, . . . , xd) ∈ Rd).

Foru ∈ L1(Rd), one sets‖u‖BV =∑d

i=1 ‖Diu‖M with, for i = 1, . . . , d, ‖Diu‖M = sup∫Rd u(x) ∂ϕ∂xi

(x)dx,ϕ ∈ C∞c (Rd), ‖ϕ‖∞ ≤ 1. One says that the functionu is in the spaceBV if u ∈ L1(Rd) and‖u‖BV < ∞.We first remark that (5.4) is true with‖u‖BV (Rd) instead of‖u‖W 1,1(Rd), and if u ∈ BV instead ofW 1,1(Rd).Indeed, to prove this result (which is classical), letρ ∈ C∞c (Rd,R+) with

∫Rd ρ(x)dx = 1. Forn ∈ N?, define

ρn = ndρ(n·). Let u ∈ BV andun = u ? ρn so that, with (5.4):

‖un‖L1? (Rd) ≤12d

d∑

i=1

‖Diun‖L1(Rd). (5.5)

But, ‖Diun‖L1(Rd) = ‖Diun‖M , and, forϕ ∈ C∞c (Rd), using Fubini’s theorem:∫

Rd

un(x)∂ϕ

∂xi(x)dx =

∫

Rd

u(x)∂

∂xi(ϕ ? ρn)(x)dx ≤ ‖Diu‖M‖ϕ‖L∞(Rd).

This leads to‖Diun‖L1(Rd) ≤ ‖Diu‖M . Sinceun → u a.e., asn →∞, at least for a subsequence, Fatou’s lemmagives, from (5.5):

‖u‖L1? (Rd) ≤12d‖u‖BV ∀u ∈ BV. (5.6)

Let nowu ∈ HM(Ω). One setsu = 0 outsideΩ so thatu ∈ L1(Rd). One has

‖u‖BV = sup∫

Rd

u(x)divϕ(x)dx, ϕ ∈ C∞c (Rd,Rd), ‖ϕ‖L∞(Rd) ≤ 1, (5.7)

with ‖ϕ‖L∞(Rd) = supdi=1 ‖ϕi‖L∞(Rd) andϕ = (ϕ1, . . . , ϕd). But, forϕ ∈ C∞c (Rd,Rd) such that‖ϕ‖L∞(Rd) ≤

1, an integration by parts on each element ofM gives (wherenσ is a normal vector toσ and γ is the (d −1)−Lebesgue measure onσ):

∫

Rd

u(x)divϕ(x)dx =∑

σ∈Eδσu

∫

σ

|ϕ · nσ|dγ(x) ≤√

d‖u‖1,1,D.

Then, one has‖u‖BV ≤√

d‖u‖1,1,D and (5.6) leads to (5.3).¤

74

5.2.2 Discrete embedding ofW 1,p(Ω) in Lp?(Ω), 1 < p < d

We now prove a discrete Sobolev embedding for1 < p < d and for meshes in sense of Definition 1.3.

Lemma 5.2 Under hypothesis (1.5), let D be a pointed strictly star-shaped polygonal finite volume space dis-cretization ofΩ in the sense of Definition 1.3. Letη > 0 such that

η ≤ dK,σ/dL,σ ≤ 1/η, ∀σ ∈ Eint, m(σ) = K,L. (5.8)

Then, there existsC1, only depending ond, p andη such that (see Definition 5.1):

‖u‖Lp? (Ω) ≤ C1‖u‖1,p,D ∀u ∈ HM(Ω), (5.9)

wherep? = pdd−p and

‖u‖p1,p,D =

∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(δσu

dσ

)p

, (5.10)

wheredσ is defined by (1.12).

Proof. Note first that definition (5.10) is compatible with (5.1) forp = 1, since in (5.10), for an internal edgeσ ∈ Eint with Mσ = K,L, we have

dK,σ

dσ+

dL,σ

dσ= 1.

We follow here also the proof of Sobolev embedding due to L. Nirenberg. Letα be such thatα1? = p? (that isα = p(d− 1)/(d− p) > 1). Let u ∈ HM(Ω). Inequality (5.3) applied with|u|α instead ofu leads to:

(∫

Ω

|u(x)|p?

dx

) d−1d

≤∑

σ∈Em(σ)δσ|u|α.

Let us remark that, for allσ ∈ Eint withMσ = K,L, we haveδσ|u|α ≤ α(|uK |α−1 + |uL|α−1)δσu, and for allσ ∈ Eext, we haveδσ|u|α ≤ α|uK |α−1δσu. This gives

(∫

Ω

|u(x)|p?

dx

) d−1d

≤ α∑

K∈M

∑

σ∈EK

m(σ)|uK |α−1δσu. (5.11)

For allσ ∈ Eint with Mσ = K, L, we can write

δσu ≤ (1 + η)dK,σδσu

dσ.

Since this also holds forσ ∈ Eext, Holder Inequality applied withq = p/(p− 1) to (5.11) yields:

(∫

Ω

|u(x)|p?

dx

) d−1d

≤ α(1 + η)

( ∑

K∈M

∑

σ∈EK

m(σ)dK,σ|uK |(α−1)q

) 1q

‖u‖1,p,D. (5.12)

Since(α− 1)q = p?, we have

∑

K∈M

∑

σ∈EK

m(σ)dK,σ|uK |(α−1)q = d

∫

Ω

|u(x)|p?

dx.

Then, noticing that(d−1)/d−1/q = 1/p?, we deduce (5.9) from (5.12) withC1 = α(1+η)d1/q only dependingond, p andη. ¤

75

5.2.3 Discrete embedding ofW 1,p(Ω) in Lq(Ω), for someq > p

Let 1 ≤ p < ∞. Lemma 5.3 gives easily a discrete embedding ofW 1,p in Lq, for someq > p, this is given in thefollowing lemma.

Lemma 5.3 Under hypothesis (1.5), let D be a pointed strictly star-shaped polygonal finite volume space dis-cretization ofΩ in the sense of Definition 1.3. Letη > 0 such that property (5.8) holds. Then, there existsq > ponly depending onp and there existsC2, only depending ond, Ω, p andη such that (see Definition 5.1):

‖u‖Lq(Ω) ≤ C2‖u‖1,p,D ∀u ∈ HM(Ω), (5.13)

where‖u‖p1,p,D is defined by (5.10).

Proof. If p = 1, one takesq = 1? and the result follows from lemma 5.1 (in this caseC2 does not depend onη).If 1 < p < d, one takesq = p? and the result is given by lemma 5.2.

If p ≥ d, one chooses anyq ∈]p,∞[ andp1 < d such thatp?1 = q (this is possible sincep?

1 tends to∞ asp1

tends tod). lemma 5.9 gives, for someC1 only depending onp, d andη, ‖u‖Lq(Ω) ≤ C1‖u‖1,p1,D. But, usingHolder inequality, there existsC3, only depending ond, p, Ω, such that‖u‖1,p1,D ≤ C3‖u‖1,p,D. Inequality (5.13)follows with C2 = C1C3. ¤

5.3 Compactness results for bounded families in discreteW 1,p(Ω) norm

Note that these properties again hold for general finite volume discretizations.

5.3.1 Compactness inLp(Ω)

We prove in this section that bounded families in the discreteW 1,p(Ω) norms are relatively compact inLp(Ω).We begin here also with the casep = 1, giving in this case a crucial inequality which holds for general polygonalpartitions ofΩ.

Lemma 5.4 Under hypothesis (1.5), letD be a polygonal finite volume space discretization ofΩ in the sense ofDefinition 1.1. Then, with the notation of Definition 5.1 :

‖u(·+ h)− u‖L1(Rd) ≤ |h|√

d‖u‖1,1,D, ∀u ∈ HM(Ω), ∀h ∈ Rd, (5.14)

whereu is defined on the wholeRd, takingu = 0 outsideΩ, and|h| is the eucllidean norm ofh ∈ Rd.

Proof. As in lemma 5.1, a proof of this result is possible with a method similar to the method for provingcompactness results for bounded families in the discreteW 1,p norms, given in [20] (and we obtain (5.14) without√

d). Indeed, this proof of [20] holds here in this case of a general partition, thanks to the fact thatp = 1. Morerestrictive assumptions are needed for the casep > 1. We give here another proof, using theBV−space, as inlemma 5.1.

Let u ∈ C∞c (Rd). Forx, h ∈ Rd, one has:

|u(x + h)− u(x)| = |∫ 1

0

∇u(x + th) · hdt| ≤ |h|∫ 1

0

|∇u(x + th)|dt.

Integrating with respect tox and using Fubini’s Theorem gives the well kown result

‖u(·+ h)− u‖L1 ≤ |h|∫

Rd

|∇u(x)|dx ≤ |h|d∑

i=1

‖Diu‖L1 , (5.15)

where∇u = (D1u, . . . , Ddu). By density ofC∞c (Rd) in W 1,1(Rd), Inequality (5.15) is also true foru ∈W 1,1(Rd).

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We proceed now as in lemma 5.1, using the same notations. Letu ∈ BV andun = u?ρn. Sinceun ∈ W 1,1(Rn),Inequality (5.15) gives, for allh ∈ Rd, ‖un(· + h) − un‖L1 ≤ |h|∑d

i=1 ‖Diun‖L1 . But, for i = 1, . . . , d, as inlemma 5.1,‖Diun‖L1 ≤ ‖Diu‖M . Then, sinceun → u in L1, asn →∞, we obtain:

‖u(·+ h)− u‖L1 ≤ |h|d∑

i=1

‖Diu‖M = |h|‖u‖BV , ∀u ∈ BV, ∀h ∈ Rd. (5.16)

Let nowu ∈ HM(Ω). One setsu = 0 outsideΩ so thatu ∈ L1(Rd). lemma 5.1 gives‖u‖BV ≤√

d‖u‖1,1,D,then:

‖u(·+ h)− u‖L1 ≤ |h|√

d‖u‖1,1,D.

¤An easy consequence of lemmas 5.1 and 5.4 is a compactness result inL1 given in the following lemma.

Lemma 5.5 Under hypothesis (1.5), let F be a family of polygonal finite volume space discretizations ofΩ in thesense of Definition 1.1. ForM ∈ F , let uM ∈ HM(Ω) and assume that there existsC ∈ R such, for allM ∈ F ,‖uM‖1,1,D ≤ C. Then the family(uM)M∈F is relatively compact inL1(Ω) and also inL1(Rd) takinguM = 0outsideΩ.

Proof. The proof is quite easy. lemma 5.1 gives that the family(uM)M∈F is bounded inL1?

(Ω). Then, sinceΩ is bounded, the family(uM)M∈F is bounded inL1(Ω) and also inL1(Rd) takinguM = 0 outsideΩ. Then,thanks to Kolmogorov Copmpactness Theorem, lemma 5.4 gives that the family(uM)M∈F is relatively compactin L1(Ω) and also inL1(Rd) takinguM = 0 outsideΩ. ¤Forp > 1, we need some an additional hypothese on the meshes, actually given by Definition 1.3 with a “uniformη”.

Lemma 5.6 Let d ≥ 1, 1 ≤ p < ∞ andΩ be a polygonal open bounded connected subset ofRd. Let F be afamily pointed strictly star-shaped polygonal finite volume space discretizations ofΩ in the sense of Definition 1.3.Let η > 0 such that the property (5.8)holds for allD ∈ F . For D ∈ F , let uD ∈ HM(Ω) and assume that thereexistsC ∈ R such, for allD ∈ F , ‖uD‖1,p,D ≤ C. Then the family(uD)D∈F is relatively compact inLp(Ω) andalso inLp(Rd) takinguD = 0 outsideΩ.

Proof. Here also the proof is quite simple. Thanks to lemma 5.3 and the fact thatΩ is bounded, the family(uD)D∈F is bounded inL1(Ω) and also inL1(Rd) taking uD = 0 outsideΩ. Thanks, once again, to the factthatΩ is bounded the family(‖uD‖1,1,D)D∈F is bounded inR. Then, as in the preceding lemma, KolmogorovCopmpactness Theorem gives that the family(‖uD‖1,1,D)D∈F is relatively compact inL1(Ω) and also inL1(Rd)takinguD = 0 outsideΩ.

In order to conclude we use, once again, lemma 5.3. It gives that the family(uD)D∈F is bounded inLq(Ω) forsomeq > p. With the relative compactness inL1(Ω), this leads to the fact that the family(uD)D∈F is relativecompact inLp(Ω) (and then also inLp(Rd) takinguD = 0 outsideΩ). ¤

5.3.2 Regularity of the limit

With the hypotheses of lemma 5.6, assume thatuD → u in Lp ash(D) → 0 (lemma 5.6 gives that this is possible,at least for subsequences of sequences of meshes whose size goes to0). We prove below thatu ∈ W 1,p

0 (Ω).

Lemma 5.7 Under hypothesis (1.5), let (Dn)n∈N be a family of pointed strictly star-shaped polygonal finite vol-ume space discretizations ofΩ in the sense of Definition 1.2. Letη > 0 such that, for alln ∈ N, the property (5.8)holds withD = Dn. For n ∈ N, let u(n) ∈ HMn(Ω) and assume that there existsC4 ∈ R such, for alln ∈ N,‖u(n)‖1,p,Dn ≤ C4. Assume also thath(Dn) → 0 asn →∞. Then:

1. There exists a subsequence of(u(n))n∈N, still denoted by(u(n))n∈N, andu ∈ Lp(Ω) such thatu(n) → u inLp(Ω) asn →∞.

77

2. u ∈ W 1,p0 (Ω) and

‖∇u‖Lp(Ω)d = ‖|∇u|‖Lp(Ω) ≤1 + η

ηd

p−1p C4. (5.17)

Proof. The fact that there exists a subsequence of(u(n))n∈N, still denoted by(u(n))n∈N, andu ∈ Lp(Ω) suchthat u(n) → u in Lp(Ω) asn → ∞ is a consequence of the relative compactness of(u(n))n∈N in Lp given inlemma 5.6. Assuming thatu(n) → u in Lp(Ω) asn →∞, we have now to prove thatu ∈ W 1,p

0 (Ω).

Lettingu(n) = 0 andu = 0 outsideΩ, one also hasu(n) → u in Lp(Rd). The method consists to construct someapproximate gradient, namely∇Dn

u(n), bounded inLp(Ω), equal to0 outsideΩ and converging, at least in thedistribution sense, to∇u.

Step 1Construction of∇Du, for u ∈ HM(Ω), and propertiesLet n ∈ N andD = Dn. For this step, one setsu = u(n) (not to be confused with the limit of the sequence(u(n))n∈N). Forσ ∈ E , one setsuσ = 0 if σ is on the boundary ofΩ. Otherwise, one hasMσ = K, L and wechoose a valueuσ betweenuK anduL. (it is possible to choose, for instance;uσ = 1

2 (uK + uL) but any otherchoice betweenuK anduL is possible). Then, one defines∇Du onK ∈ D on the following way:

∇Du =1

m(K)

∑

σ∈EK

m(σ)nK,σ(uσ − uK).

The function∇Du is constant on eachK ∈M and, onK, using Holder Inequality:

|∇Du|p ≤ 1(m(K))p

(∑

σ∈EK

m(σ)nK,σ(uσ − uK))p ≤ 1(m(K))p

(∑

σ∈EK

m(σ)dK,σ)p−1∑

σ∈EK

m(σ)dK,σ(δσu

dK,σ)p.

Since∑

σ∈EKm(σ)dK,σ = dm(K), one deduces

|∇Du|p ≤ dp−1

m(K)

∑

σ∈EK

m(σ)dK,σ(δσu

dK,σ)p.

This gives aLp− estimate on∇Du in (Lp(Ω))d (or in (Lp(Rd))d, setting∇Du = 0 outsideΩ), in terms of‖u‖1,p,D, namely:

‖|∇Du|‖Lp ≤ 1 + η

ηd

p−1p ‖u‖1,p,D. (5.18)

In order to prove, in the next step, the convergence of this approximate gradient, we compute now the integral ofthis gradient against a test function. Letϕ ∈ C∞c (Rd;Rd), ϕK the mean value ofϕ on K ∈ D andϕσ the meanvalue ofϕ onσ. Then:

∫

Rd

∇Du · ϕdx =∑

K∈D

∑

σ∈EK

m(σ)nK,σ(uσ − uK)ϕK =∑

K∈D

∑

σ∈EK

m(σ)nK,σ(−uK)ϕσ + R(u, ϕ), (5.19)

withR(u, ϕ) =

∑

K∈D

∑

σ∈EK

m(σ)nK,σ(uσ − uK)(ϕK − ϕσ).

Then there existsCϕ only depending onϕ, d, p andΩ such that|R(u, ϕ)| ≤ Cϕh(D)‖u‖1,p,D. Equation (5.19)can also be written as:

∫

Rd

∇Du · ϕdx =∑

K∈D

∫

K

(−uK)div(ϕ)dx + R(u, ϕ) = −∫

Rd

udiv(ϕ)dx + R(u, ϕ). (5.20)

Step 2Convergence of∇Dnu(n) to∇u and proof ofu ∈ W 1,p0 (Ω) .

78

We consider now the sequence(u(n))n∈N. Inequality (5.18) gives

‖|∇Du(n)|‖Lp ≤ 1 + η

ηd

p−1p ‖u(n)‖1,p,D.

Then, the sequence(∇Du(n))n∈N is bounded inLp(Rd)d and we can assume, up to a subsequence, that∇Du(n)

converges to somew weakly inLp(Rd)d, asn →∞ and‖|w|‖Lp ≤ 1+ηη d

p−1p C.

Let ϕ ∈ C∞c (Rd;Rd), Equation (5.20) gives∫

Rd

∇Du(n) · ϕdx = −∫

Rd

u(n)div(ϕ)dx + R(u(n), ϕ). (5.21)

Thanks to|R(u(n), ϕ)| ≤ Cϕh(Dn)‖u(n)‖1,p,Dn , one hasR(u(n), ϕ) → 0, asn →∞. Sinceu(n) → u in Lp(Rd)asn →∞, passing to the limit in (5.21) gives:

∫

Rd

w · ϕdx = −∫

Rd

udiv(ϕ)dx.

Sinceϕ is arbitrary inC∞c (Rd,Rd), one deduces that∇u = w. Thenu ∈ W 1,p(Rd) and‖|∇u|‖Lp ≤ 1+ηη d

p−1p C.

Finally, sinceu = 0 outsideΩ, one hasu ∈ W 1,p0 (Ω). ¤

5.4 Properties in the case of the∆−adapted discretizations

We now focus on some properties which use the orthogonality property assumed in Definition 1.4.Let us first give the two following results, already proven in [20]. Nevertheless, since their proof is quite short, weprovide it for the sake of completeness, and in order to make clear the differences with the general framework ofpointed finite volume discretizations.

Lemma 5.8 Under hypothesis (1.5), letD be a pointed∆−adapted polygonal finite volume space discretizationof Ω in the sense of Definition 1.4. We prolong all the elements ofHM(Ω) by0 in Rd \ Ω. Then

‖u(·+ h)− u‖2L2(Rd) ≤ |h|(|h|+ 4h(M))‖u‖21,2,D, ∀u ∈ HM(Ω). (5.22)

Proof. Forσ ∈ E , defineχσ fromRd × Rd to 0, 1 by

1. χσ(x, y) = 1 if ]x,y[∩σ 6= ∅ and there existsz ∈]x, y[∩σ such that[x,z] ⊂ Ω or [z, y] ⊂ Ω,

2. χσ(x, y) = 0 otherwise.

Let h ∈ Rd, h 6= 0. One has

|u(x + h)− u(x)| ≤∑

σ∈Eχσ(x, x + h)δσu =

∑

K∈M

∑

σ∈EK

χσ(x,x + h)dK,σδσu

dσ, for a.e.x ∈ Ω

(see Definition (5.2) for the definition ofδσu).This gives, using the Cauchy-Schwarz inequality,

|u(x+h)−u(x)|2 ≤∑

K∈M

∑

σ∈EK

dK,σχσ(x,x+h)(δσu)2

d2σcK,σ

∑

K∈M

∑

σ∈EK

χσ(x, x+h)dK,σcK,σ, for a.e.x ∈ Rd,

(5.23)wherecK,σ = |nK,σ · h

|h| |.Let us now prove that

79

∑

K∈M

∑

σ∈EK

χσ(x, x + h)dK,σcK,σ ≤ |h|+ 4 h(M), (5.24)

for a.e.x ∈ Rd.Let x ∈ Rd such thatσ ∩ [x, x + h] contains at most one point, for allσ ∈ E , and[x, x + h] does not contain anyvertex ofM (proving (5.25) for such pointsx gives (5.25) for a.e.x ∈ Rd, sinceh is fixed).Let y, z ∈ [x,x + h] such that(z − y) · h > 0 and [y, z] ⊂ Ω. Let us assume that there existsσ ∈ Eint

such thatχσ(y,z) = 1. We then consider the sequence(Ki), i = 1, . . . , k of the elements ofM such that]y, z[∩Ki 6= ∅, y ∈ K1, z ∈ Kk, and for anyσ ∈ E such thatχσ(y, z) = 1, there existsi = 1, . . . k − 1 such

thatMσ = Ki,Ki+1. Definition 1.4 implies thatnKi,σ =xKi+1−xKi

dσandnKi+1,σ = −nKi,σ. Thanks to

(z − y) · h > 0, we have

nKi,σ · h =xKi+1 − xKi

dσ· h ≥ 0.

Hence we get

∑

K∈M

∑

σ∈EK

χσ(y, z)dK,σcσ =k−1∑

i=1

(xKi+1 − xKi) · h

|h| = (xKk− xK1) ·

h

|h| .

Since|y − xK1 | ≤ h(M) and|z − xKk| ≤ h(M), this leads to

∑

K∈M

∑

σ∈EK

χσ(y, z)dK,σcσ ≤ |y − z|+ 2 h(M). (5.25)

In the case where there does not existσ ∈ Eint such thatχσ(y,z) = 1, (5.25) remains true. Note that (5.25)implies (5.24) if[x, x + h] ⊂ Ω.We now consider the case where[x,x+h] 6⊂ Ω. In the case wherex /∈ Ω andx+h /∈ Ω, thenχσ(x,x+h) = 0for all σ ∈ E , and (5.24) holds. In the case wherex ∈ Ω andx + h /∈ Ω, there existsz ∈ [x, x + h] ∩ ∂Ω suchthat [x,z] ⊂ Ω. We then haveχσ(x, x + h) = χσ(x, z) for all σ ∈ E . Then (5.25), withy = x, implies (5.24).In the case wherex /∈ Ω andx + h ∈ Ω, there existsy ∈ [x,x + h] ∩ ∂Ω such that[y, x + h] ⊂ Ω. We thenhaveχσ(x,x + h) = χσ(y, x + h) for all σ ∈ E . Then (5.25), withz = x + h, implies (5.24).Finally, in the case wherex ∈ Ω andx + h ∈ Ω, there existsz ∈ [x, x + h] ∩ ∂Ω such that[x, z] ⊂ Ω andy ∈ [x, x + h] ∩ ∂Ω such that[y, x + h] ⊂ Ω. We then haveχσ(x, x + h) = χσ(x, z) + χσ(y, x + h) for allσ ∈ E , and we get from (5.25):

∑

K∈M

∑

σ∈EK

χσ(x,x + h)dK,σcσ ≤ |x− z|+ 2 h(M) + |x + h− y|+ 2 h(M),

which implies (5.24).In order to conclude the proof of Lemma 5.8, remark that, for allK ∈M andσ ∈ EK ,

∫

Rd

χσ(x, x + h)dx ≤ m(σ)cK,σ|h|.

Therefore, integrating (5.23) overRd yields, with (5.24),

‖u(·+ h)− u‖2L2(Rd) ≤( ∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(δσu

dσ

)2)|h|(|h|+ 4 h(M)),

which is (5.22).¤Applying the previous lemma with anyh ∈ Rd such that|h| = diam(Ω) yields the following one.

Lemma 5.9 (Discrete Poincare inequality) Under hypothesis (1.5), let D be a pointed∆−adapted polygonalfinite volume space discretization ofΩ in the sense of Definition 1.4. Then the following inequality holds:

‖u‖L2(Ω) ≤ (diam(Ω) + 2 h(M))‖u‖1,2,D ≤ 3 diam(Ω)‖u‖1,2,D, ∀u ∈ HM(Ω). (5.26)

80

Remark 5.1 Lemma 5.2 provides a similar inequality to (5.26), but the constantC1 given in this lemma dependson some regularity factor of the discretization which is not required here. Note that in [20], under additionalhypotheses on the pointsxK for anyK ∈M such thatEK ∩ Eext 6= ∅, the inequality (5.26)can be replaced by

‖u‖L2(Ω) ≤ diam(Ω)‖u‖1,2,D, ∀u ∈ HM(Ω).

The next lemma provides the weak convergence of a discrete gradient (this lemma has been first given in [24]),defined by constant values by coneDK,σ.

Lemma 5.10 (Weak convergence of a discrete gradient)Under hypothesis (1.5), let F be a family of pointed∆−adapted polygonal finite volume space discretization ofΩ in the sense of Definition 1.4.We assume that, for allD ∈ F , there existsuD ∈ HM(Ω) such that‖uD‖1,2,D remains bounded by a valueC5,and such that there existsu ∈ L2(Ω) such thatuD → u in L2(Ω). We again define, for allu ∈ HM(Ω)

δK,σu = uL − uK , ∀σ ∈ Eint, Mσ = K,L,δK,σu = −uK , ∀σ ∈ Eext, Mσ = K. (5.27)

For anyD ∈ F , we define the functionGD ∈ LD(Ω)d (see Definition 1.3), by

GDK,σ = d

δK,σu

dσnK,σ, ∀K ∈M, ∀σ ∈ EK . (5.28)

Thenu ∈ H10 (Ω) holds, andGD weakly converges to∇u in L2(Ω)d ash(M) → 0.

Proof. We first prolongu andGD by 0 outsideΩ. Let us first remark that, thanks to the Cauchy-Schwarzinequality and usingm(DK,σ) = m(σ)dK,σ/d,

‖GD‖2L2(Rd)d = d∑

K∈M

∑

σ∈EK

m(σ)dK,σ

(δσu

dσ

)2

= d‖uD‖21,2,D.

Since‖GD‖L2(Rd)d remains bounded, we can extract fromF a subfamily, again denotedF , such that there existsG ∈ L2(Rd)d andGD weakly converges inL2(Rd)d to ∇u in G ash(M) → 0. Let us prove thatG = ∇u,which implies, by uniqueness of the limit, that the whole family converges to∇u ash(M) → 0.Let ϕ ∈ C1

c (Ω)d be given. Let us denote byϕD ∈ LD(Ω)d the element defined by the value1m(σ)

∫σ

ϕ(x)dγ(x),in DK,σ for all K ∈Mσ, again prolonged by0 outsideΩ. We then have thatϕD converges toϕ in L2(Rd)d. Weget, by reordering the terms in the summation, that

∫

Rd

GD(x) ·ϕ(x)dx = −∑

K∈MuK

∑

σ∈EK

∫

σ

ϕ(x)dγ(x)nK,σ = −∫

Rd

uD(x)div(ϕ)(x)dx,

which implies, by passing to the limith(M) → 0,∫

Rd

G(x) ·ϕ(x)dx = −∫

Rd

u(x)div(ϕ)(x)dx.

This proves thatu ∈ H1(Rd). Sinceu = 0 outsideΩ, we get thatu ∈ H10 (Rd), and that

∫

Rd

G(x) ·ϕ(x)dx =∫

Rd

∇u(x) ·ϕ(x)dx,

which proves thatG = ∇u, hence completing the proof of the lemma.¤We can now state a compactness lemma, which holds without the regularity hypothesis on the mesh (5.8).

Lemma 5.11 (Compactness inL2(Ω)) Under hypothesis (1.5), let F be a family of pointed∆−adapted polyg-onal finite volume space discretization ofΩ in the sense of Definition 1.4. ForM ∈ F , let uM ∈ HM(Ω) andassume that there existsC ∈ R such, for allM ∈ F , ‖uM‖1,2,D ≤ C. Then the family(uM)M∈F is relativelycompact inL2(Ω) and also inL2(Rd) takinguM = 0 outsideΩ. Moreover, the limitu ∈ L2(Ω) of any convergingsequence extracted from the family ash(M) → 0 is such thatu ∈ H1

0 (Ω).

81

Proof. The proof is quite easy. lemma 5.9 gives that the family(uM)M∈F is bounded inL2(Ω) and inL2(Rd)takinguM = 0 outsideΩ. Thanks to en, thanks to Kolmogorov Copmpactness Theorem, lemma 5.4 gives that thefamily (uM)M∈F is relatively compact inL2(Ω) and also inL2(Rd). The regularity of the limit is an immediateconsequence of Lemma 5.10. Note that it can also be proven by passing to the limith(M) → 0 in (5.22) (this isthe method used in [20]).¤

Lemma 5.12 (Strong convergence of an interpolated approximate gradient)Under hypothesis (1.5), letF bea family of pointed∆−adapted polygonal finite volume space discretization ofΩ in the sense of Definition 1.4.Letϕ ∈ C2

c (Ω) be given.For anyD ∈ F , we define the functionHD ∈ LD(Ω)d (see Definition 1.3), by

HDK,σ =

δK,σΠMϕ

dσnK,σ +∇ϕ(xK)− (∇ϕ(xK) · nK,σ)nK,σ, ∀K ∈M, ∀σ ∈ EK , (5.29)

which can also be developped in

HDK,σ =

ϕ(xL)− ϕ(xK)dσ

nK,σ +∇ϕ(xK)− (∇ϕ(xK) · nK,σ)nK,σ,

∀K ∈M, ∀σ ∈ EK ∩ Eint, Mσ = K, LHD

K,σ =0− ϕ(xK)

dσnK,σ +∇ϕ(xK)− (∇ϕ(xK) · nK,σ)nK,σ,

∀K ∈M, ∀σ ∈ EK ∩ Eext.

ThenHD converges to∇ϕ in L∞(Ω)d ash(M) → 0.

Proof. We have, forK ∈M andσ ∈ EK ∩ Eint with Mσ = K, L,

ϕ(xL)− ϕ(xK) = dσ∇ϕ(xK) · nK,σ +d2

σ

2RK,σ,

with |RK,σ| is bounded by someL∞ norm of the second order partial derivatives ofϕ, denoted byC6. We considerh(M) small enough, such thatϕ vanishes on allK ∈M such thatEK ∩ Eext 6= ∅, hence forσ ∈ EK ∩ Eext,

HDK,σ = 0 = ∇ϕ(xK).

We then get

|HDK,σ −∇ϕ(xK)| ≤ dσ

2C6 ≤ h(M)C6,∀K ∈M, ∀σ ∈ EK .

This concludes the proof of the lemma.¤Let us now give the following lemma [27; 18].

Lemma 5.13 (Convergence of discrete scalar product,∆−adapted mesh)Under hypothesis (1.5), let F be afamily of pointed∆−adapted polygonal finite volume space discretization ofΩ in the sense of Definition 1.4.We assume that, for allD ∈ F , there existsuD ∈ HM(Ω) such that‖uD‖1,2,D remains bounded by a valueC7,and such that there existsu ∈ H1

0 (Ω) such thatuD → u in L2(Ω). We also assume that, for allD ∈ F , thereexistsgD ∈ LM(Ω), such that there existsg ∈ L2(Ω) with gD → g in L2(Ω).For anyD ∈ F , we define

[v, w]D,α =∑

K∈M

∑

σ∈EK

αK,σm(σ)dK,σδK,σv

dσ

δK,σw

dσ, ∀v, w ∈ HM(Ω), ∀α ∈ LD(Ω). (5.30)

Then the following holds:

limh(M)→0

[uD,ΠMϕ]D,gD =∫

Ω

g(x)∇u(x) · ∇ϕ(x)dx, ∀ϕ ∈ C2c (Ω). (5.31)

Moreover, we have, ifg ∈ L∞(Ω),

lim infh(M)→0

[uD, uD]D,|gD| ≥∫

Ω

|g(x)||∇u(x)|2dx. (5.32)

82

Remark 5.2 Under the hypotheses of the lemma, one can derive the existence ofu from results obtained in Section5.2.

Proof. Let us first show (5.31). LetGD ∈ LD(Ω)d be defined by (5.28) andHD ∈ LD(Ω)d be defined by (5.29).We deduce from Lemma 5.12, thatgDHD tends tog∇ϕ in L2(Ω), which implies, by weak/strong convergence,that

limh(M)→0

∫

Ω

gD(x)GD(x) ·HD(x)dx =∫

Ω

g(x)∇u(x) · ∇ϕ(x)dx.

We now remark that, thanks to definition (5.30) of[·, ·]D,·,∫

Ω

gD(x)GD(x) ·HD(x)dx = [uD,ΠMϕ]D,gD ,

which concludes the proof of (5.31).Let us now prove (5.32). We now assume thatg ∈ L∞(Ω). We have

[uD −ΠMϕ, uD −ΠMϕ]D,|gD| ≥ 0, ∀D ∈ F .

Hence we get[uD, uD]D,|gD| ≥ 2[uD, ΠMϕ]D,|gD| − [ΠMϕ,ΠMϕ]D,|gD|, ∀D ∈ F .

Applying (5.31), we get

limh(M)→0

[uD,ΠMϕ]D,gD =∫

Ω

|g(x)|∇u(x) · ∇ϕ(x)dx.

Using that‖ΠMϕ‖1,1,D remains bounded, we can again apply (5.31), substitutingΠMϕ to uD. It leads to

limh(M)→0

[ΠMϕ,ΠMϕ]D,gD =∫

Ω

|g(x)|∇ϕ(x) · ∇ϕ(x)dx.

Hence we get

lim infh(M)→0

[uD, uD]D,|gD| ≥∫

Ω

|g(x)|(2∇ϕ(x)−∇u(x)) · ∇ϕ(x)dx.

Since the above inequality holds for allϕ ∈ C2c (Ω), lettingϕ → u in H1(Ω) provides the convergence inL1(Ω)

of (2∇ϕ−∇u) · ∇ϕ to∇u. Hence we get (5.32).¤In the same spirit as the previous lemma, let us give the following result, given in [30].

Lemma 5.14 (Weak-Strong convergence for space-time problems)Under hypothesis (1.5), let T > 0 be given. Let(Dm)m∈N be a sequence of finite volume discretizations ofΩ × (0, T ) in the sense of Definitions 1.4 and 1.7 such thatlim

m→+∞size(Dm) = 0 and such that the regularity

property

∃θ ∈ R+ such that∀K ∈Mm,∑

L∈NK

mK|LdK|L ≤ θm(K). (5.33)

is satisfied. Let(vDm)m∈N ⊂ L2(Ω × (0, T )) (resp. (wDm)m∈N ⊂ L2(Ω × (0, T ))) be a sequence of piecewiseconstant functions corresponding to a sequence of discrete functions(vDm)m∈N (resp. (wDm)m∈N) fromMm ×[[0, Nm + 1]] toR. Assume that there exists a real valueC8 > 0 independent onm verifying

N∑n=0

δtn∑

K∈M

∑

L∈NK

m(K|L)dK|L

(vn+1L − vn+1

K )2 ≤ C8,

83

and that the sequence(vDm)m∈N (resp. (wDm)m∈N) converges to some functionv ∈ L2(Ω × (0, T )) (resp.w ∈ L2(Ω × (0, T ))) weakly (resp. strongly) inL2(Ω × (0, T )), asm → +∞. Letϕ ∈ C∞(Rd × R) such thatϕ(·, T ) = 0 and∇ϕ · n = 0 on∂Ω× (0, T ). For all m ∈ N, let Am be defined by

Am = −N∑

n=0

δtn∑

K∈Mϕ(xK , tn+1)

∑

L∈NK

m(K|L)dK|L

wn+1K,L (vn+1

L − vn+1K ),

where, for all(K, L) ∈ E andn ∈ [[0, N ]], wn+1K,L = wn+1

L,K , andwn+1K,L is either equal town+1

K or to wn+1L . Then

v ∈ L2(0, T ;H1(Ω)) and

limm→+∞

Am =∫ T

0

∫

Ω

w(x, t)∇v(x, t)∇ϕ(x, t)dxdt.

5.5 Lemma used for time translates

Lemma 5.15 Let (tn)n∈Z be a stricly increasing sequence of real values such thatδtn := tn+1 − tn is uniformlybounded, lim

n→−∞tn = −∞ and lim

n→∞tn = ∞. For all t ∈ R, we denote byn(t) the elementn ∈ Z such that

t ∈ [tn, tn+1). Let(an)n∈Z be a family of non negative real values with a finite number of non zero values. Then

∫

R

n(t+τ)∑

n=n(t)+1

(δtnan+1)dt = τ∑

n∈Z(δtnan+1), ∀τ ∈ (0, +∞), (5.34)

and

∫

R

n(t+τ)∑

n=n(t)+1

δtn

an(t+ζ)+1dt ≤ (τ + max

n∈Zδtn)

∑

n∈Z(δtnan+1), ∀τ ∈ (0, +∞), ∀ζ ∈ R. (5.35)

Proof. Let us define the functionχ(t, n, τ) by χ(t, n, τ) = 1 if t < tn andt + τ ≥ tn, elseχ(t, n, τ) = 0. Wehave ∫

R

n(t+τ)∑

n=n(t)+1

(δtnan+1)dt =∫

R

∑

n∈Z(δtnan+1χ(t, n, τ))dt =

∑

n∈Z

(δtnan+1

∫

Rχ(t, n, τ)dt

).

Since∫R χ(t, n, τ)dt =

∫ tn

tn−τdt = τ , thus (5.34) is proven.

We now turn to the proof of (5.35). We define the functionχ(n, t) by χ(n, t) = 1 if n(t) = n, elseχ(n, t) = 0.We have ∫

R

n(t+τ)∑

n=n(t)+1

δtn

an(t+ζ)+1dt =

∫

R

n(t+τ)∑

n=n(t)+1

δtn

∑

m∈Zam+1χ(m, t + ζ)dt,

which yields

∫

R

n(t+τ)∑

n=n(t)+1

δtn

an(t+ζ)+1dt =

∑

m∈Zam+1

∫ tm+1−ζ

tm−ζ

n(t+τ)∑

n=n(t)+1

δtn

dt. (5.36)

Since we haven(t+τ)∑

n=n(t)+1

δtn =∑

n∈Z, t<tn≤t+τ

(tn+1 − tn) ≤ τ + maxn∈Z

δtn,

we can write from (5.36)

∫

R

n(t+τ)∑

n=n(t)+1

δtn

an(t+ζ)+1dt ≤ (τ + max

n∈Zδtn)

∑

m∈Zam+1

∫ tm+1−ζ

tm−ζ

dt = (τ + maxn∈Z

δtn)∑

m∈Zam+1δtm,

84

which is exactly (5.35).¤We have the following corollary of Kolmogorov’s theorem.

Corollary 5.1 (Consequence of Kolmogorov’s theorem)LetΩ be a bounded open subset ofRd, d ∈ N∗, and let(wn)n∈N be a sequence of functions such that:

for all n ∈ N, wn ∈ L∞(Ω) and there exists a real valueCb > 0 which does not depend onn such that‖wn‖L∞(Ω) ≤ Cb,

there exists a real valueCK > 0 and a sequence of non negative real values(µn)n∈N verifying limn→∞

µn = 0and

∫

Ωξ

(wn(x + ξ)− wn(x))2dx ≤ CK(|ξ|+ µn), ∀n ∈ N, ∀ξ ∈ Rd.

where, for allξ ∈ Rd, we setΩξ = x ∈ Rd, [x, x + ξ] ⊂ Ω.

Then there exists a subsequence of(wn)n∈N, again denoted(wn)n∈N, and a functionw ∈ L∞(Ω) such thatwn → w in Lp(Ω) asn →∞ for all p ≥ 1.

Proof. We first extend the definition ofwn onRd by the value0 onRd \ Ω. We then prove that

lim|ξ|→0

supn∈N

∫

Rd

(wn(x + ξ)− wn(x))2dx = 0. (5.37)

Indeed, letε > 0. Since∪η>0x ∈ Ω, B(x, η) ⊂ Ω = Ω, then there existsη > 0 such that

meas(Ω \ x ∈ Ω, B(x, η) ⊂ Ω) ≤ ε.

Thus, for allξ ∈ Rd verifying |ξ| ≤ η, we have

x ∈ Ω, B(x, η) ⊂ Ω ⊂ Ωξ ∪ Ω−ξ ⊂ Ω

and therefore meas(Ω \ (Ωξ ∪ Ω−ξ)) ≤ ε. Let n0 ∈ N such that, for alln > n0, µn ≤ ε. Thanks to the theoremof continuity in means, for alln = 0, . . . , n0, there existsηn > 0 such that, for allξ ∈ Rd verifying |ξ| ≤ ηn, wehave

∫Rd(wn(x + ξ)− wn(x))2dx ≤ ε.

We now takeξ ∈ Rd verifying |ξ| ≤ min(η, (ηn)n=0,...,n0 , ε). We then get that, for alln = 0, . . . , n0, theinequality

∫Rd(wn(x + ξ)− wn(x))2dx ≤ ε holds, and for alln ∈ N such thatn > n0, then

∫

Rd

(wn(x + ξ)− wn(x))2dx ≤ 4C2b ε +

∫

Ωξ

(wn(x + ξ)− wn(x))2dx,

and ∫

Ωξ

(wn(x + ξ)− wn(x))2dx ≤ CK(|ξ|+ µn) ≤ 2CKε.

Gathering the previous results gives (5.37). Then applying Kolmogorov’s theorem gives the conclusion of Corol-lary 5.1.¤Let us now give the following corollary of Ascoli’s theorem, useful for convergence at any time.

Theorem 5.2 Let T > 0 be given, letE be a separable Banach space andE′ be its dual. Let(un)n∈N be asequence of applications from[0, T ] to E′ such that there existsC > 0 with ‖un(t)‖E′ ≤ C for all n ∈ N andt ∈ [0, T ]. We assume that there exists a supspaceF ⊂ E with the following properties:

1. F is dense inE

85

2. for all ϕ ∈ F , the family(un,ϕ)n∈N of functionsun,ϕ : [0, T ] → R defined for allt ∈ [0, T ] byun,ϕ(t) =〈un(t), ϕ〉 is equicontinuous.

Then there exists a subsequence of(un)n∈N, again denoted(un)n∈N, and an elementu ∈ C0(0, T ; E′) (E′ beingembedded with the weak-? topology) such that for allt ∈ [0, T ], the sequence(un(t))n∈N converges tou for theweak-? topology ofE′.

Proof. Let (tm)m∈N be a sequence of elements of[0, T ], dense in[0, T ]. For all m ∈ N, the sequence(un(tm))n∈N is bounded inE′. Therefore there exists a subsequence of(un(tm))n∈N which converges for theweak-? topology of E′. Using the diagonal process, we extract from(un)n∈N a subsequence, again denoted(un)n∈N, such that for allm ∈ N, the sequence(un(tm))n∈N converges for the weak-? topology ofE′.

Let t ∈ [0, T ]. For ϕ ∈ F , let us prove that the sequence(〈un(t), ϕ〉)n∈N is a Cauchy sequence. Letε > 0 begiven. Letη > 0 be such that, for allt′ ∈ [0, T ] with |t − t′| ≤ η and for alln ∈ N, |〈un(t) − un(t′), ϕ〉| ≤ ε.Let m ∈ N be such that|t − tm| ≤ η. Since the sequence(un(tm))n∈N converges for the weak-? topology ofE′, the sequence(〈un(tm), ϕ〉)n∈N is a Cauchy sequence. Hence there existsn0 ∈ N such that, for alln, p ≥ n0,|〈un(tm)− up(tm), ϕ〉| ≤ ε. For suchn, p ≥ n0, we get

|〈un(t)− up(t), ϕ〉| ≤ |〈un(t)− un(tm), ϕ〉|+ |〈un(tm)− up(tm), ϕ〉|+ |〈up(tm)− up(t), ϕ〉| ≤ 3 ε.

This shows that the sequence(〈un(t), ϕ〉)n∈N is a Cauchy sequence.

Let ψ ∈ E. Let us prove that the sequence(〈un(t), ψ〉)n∈N is a Cauchy sequence. Letε > 0 be given. We firstchooseϕ ∈ F such that‖ψ − ϕ‖E ≤ ε. Since the sequence(〈un(t), ϕ〉)n∈N is a Cauchy sequence, there existsn0 ∈ N such that, for alln, p ≥ n0, |〈un(t)− up(t), ϕ〉| ≤ ε. For suchn, p ≥ n0, we get

|〈un(t)− up(t), ψ〉| ≤ |〈un(t), ψ − ϕ〉|+ |〈un(t)− up(t), ϕ〉|+ |〈up(t), ϕ− ψ〉| ≤ (2 C + 1)ε.

This shows that the sequence(〈un(t), ψ〉)n∈N is a Cauchy sequence. Since for allt ∈ [0, T ], n ∈ N andψ ∈ E,we have|〈un(t), ψ〉| ≤ C‖ψ‖E , we get that

∣∣∣ limn→∞

〈un(t), ψ〉∣∣∣ ≤ C‖ψ‖E .

Hence the applicationu defined, for allt ∈ [0, T ] by u(t)(ψ) = limn→∞〈un(t), ψ〉 for all ψ ∈ E is such thatu(t) ∈ E′.

Let us show thatu ∈ C0(0, T ; E′), E′ being embedded with the weak-? topology. Lett ∈ [0, T ] andψ ∈ E. Letε > 0 be given. We first chooseϕ ∈ F such that‖ψ − ϕ‖E ≤ ε. Let η > 0 be such that, for allt′ ∈ [0, T ] with|t−t′| ≤ η and for alln ∈ N, |〈un(t)−un(t′), ϕ〉| ≤ ε. Passing to the limitn →∞, we get|〈u(t)−u(t′), ϕ〉| ≤ ε.We then get

|〈u(t)− u(t′), ψ〉| ≤ |〈u(t), ψ − ϕ〉|+ |〈u(t)− u(t′), ϕ〉|+ |〈u(t′), ϕ− ψ〉| ≤ (2 C + 1)ε.

This shows that the functionuψ : [0, T ] → R defined for allt ∈ [0, T ] by uψ(t) = 〈u(t), ψ〉 is continuous, henceprovingu ∈ C0(0, T ; E′).¤

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Index

notation|D| Def. 1.7, 8δK,σu (2.17), 11δn+1K,σ g(u) (3.9), 36

δσu (5.2), Def. 5.1, 72δn+1σ g(u) (3.30), Lemma 3.5, 43

∂K Def. 1.1, 6DK,σ (1.11), Def. 1.3, 7dK,σ Def. 1.2, 6Dσ Def. 1.3, 7dσ (1.12), Def. 1.3, 7δtn Def. 1.6, 8E Def. 1.1, 6EK Def. 1.1, 6∇Du (2.30), 14h(M) (1.7), Def. 1.1, 6HM(Ω) Def. 1.1, 6LD(Ω) Def. 1.3, 7LD,0(Ω) 4.2, 61M Def. 1.1, 6m(K) Def. 1.1, 6m(σ) Def. 1.1, 6Mσ Def. 1.1, 6NK Def. 1.1, 6nK,σ Def. 1.1, 6‖u‖1,1,D (5.1), Definition 5.1, 72‖u‖BV (5.7), 73P Def. 1.2, 6PE 4.2, 61PD (1.9), Def. 1.2, 6PM (1.8), Def. 1.2, 6ΠMu Def. 1.1, 6[v, w]D,α (5.30), Lemma 5.13, 81un+1

K,σ (3.8), 36XD Def. 1.1, 6XD,0 Def. 1.1, 6XE 4.2, 61xK Def. 1.2, 6xσ Def. 1.2, 6YD (2.47), 17

discretization

∆−adapted pointed polygonal finite volume spacediscretization, 7

∆−super-adapted pointed polygonal finite volumespace discretization, 8

Pointed polygonal finite volume space discretiza-tion, 6

Pointed star-shaped polygonal finite volume spacediscretization, 7

Space-time discretization, 8Time discretization, 8

Polygonal finite volume space discretization, 6

Star shaped, 7

Topological degree, 72

88