Finite volume schemes for the <span class="mathjax-formula">$p$</span>-laplacian on cartesian meshes

Vol. 38, N^o 6, 2004, pp. 931–959 DOI: 10.1051/m2an:2004045

FINITE VOLUME SCHEMES FOR THE P-LAPLACIAN ON CARTESIAN MESHES

Boris Andreianov

, Franck Boyer

and Florence Hubert

Abstract. This paper is concerned with the finite volume approximation of the p-Laplacian equation with homogeneous Dirichlet boundary conditions on rectangular meshes. A reconstruction of the norm of the gradient on the mesh’s interfaces is needed in order to discretize the p-Laplacian operator. We give a detailed description of the possible nine points schemes ensuring that the solution of the resulting finite dimensional nonlinear system exists and is unique. These schemes, called admissible, are locally conservative and in addition derive from the minimization of a strictly convexe and coercive discrete functional. The convergence rate is analyzed when the solution lies in W^2,p. Numerical results are given in order to compare different admissible and non-admissible schemes.

Mathematics Subject Classification. 35J65, 65N15, 74S10.

Received: July 12, 2004.

1. Introduction

In this paper, we are interested in the study of the numerical approximation of solutions to the 2D p-Laplacian with homogeneous Dirichlet boundary conditions (1< p <+∞):

−div

|∇u|^p−2∇u

=f, on Ω,

u= 0, on∂Ω. (1)

This equation arises as a model problem in various physical situations such as non-Newtonian flows (for example in glaciology models [14, 15]), turbulent flows, or in the study of flows in porous media. Some of these models are described and studied for instance in [8].

Throughout this paper, we will denote byp = _p−1^p the conjugate exponent ofp. Let us recall that for any data f ∈W^−1,p(Ω), problem (1) is well-posed inW₀^1,p(Ω). This can be proved easily using the monotony of the p-Laplacian operator or equivalently using the fact that (1) is the Euler-Lagrange equation for the problem

Keywords and phrases. Finite volume methods, p-Laplacian, error estimates.

1 D´epartement de Math´ematiques, Universit´e de Franche-Comt´e, 16 route de Gray, 25030 Besan¸con Cedex, France.

2 Laboratoire d’Analyse, Topologie et Probabilit´es, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France.

e-mail:fboyer@cmi.univ-mrs.fr

c EDP Sciences, SMAI 2004

of minimizing the coercive and strictly convex functional J :W₀^1,p(Ω) →R

u→ 1 p

Ω|∇u(z)|^pdz−

Ωf(z)u(z) dz (2)

on the whole spaceW₀^1,p(Ω). Formula (2) is only valid forf ∈L^p(Ω) but we may replace the last integral by a duality bracket iff is inW^−1,p(Ω). Furthermore, we have thea prioriestimate

u_W^1,p≤ f_W^p−11−1,p ≤Cf_L^p−1p¹ . (3) Many authors (see for instance [4–6, 13]) have studied the finite elements approximation of (1). In this work, we propose a new approach in the framework of cell-centered finite volume schemes. Indeed, the choice of finite volume methods is natural as soon as the equation we consider is in a divergence form. It consists, for this kind of problems, in approaching in a conservative way, the equation expressing the balance of fluxes in each control volume.

It appears that the approximation of the fluxes for (1) is not straightforward. Indeed, the difficulty arises from the need to approximate the complete gradient of the solution (and not only its normal derivatives on the edges as for the Laplace equation on admissible meshes). This problem of the complete gradient approximation for finite volume methods has interested many authors in the last few years in different contexts. We refer for instance to [7, 11, 12] for anisotropic diffusion problems and to [9] for the Laplace equation on almost arbitrary meshes. In the context of non-linear diffusion problems, the finite volume approach with gradient reconstruction was only considered, to our knowledge, in the work by Andreianov, Gutnic and Wittbold in [3], for general elliptic-parabolic equations and systems involving the p-Laplacian. Using an original “continuous”

approach, these authors obtain a convergence result for general schemes that inherit the essential properties of the underlying continuous problem; they are not interested in the convergence rate. Their examples of admissible schemes include meshes dual to triangular ones, but not arbitrary rectangular meshes.

In this paper, we provide a detailed study of the approximation of the model elliptic problem (1) on general rectangular meshes. We first give a complete description of the possible nine points finite volume schemes for which existence and uniqueness of the approximate solution is ensured in spite of the non-linearity of the problem. Moreover, this approximate solution can be computed by solving a finite dimensional minimization problem. Then we analyze the convergence properties of these schemes and obtain error estimates. We finally propose numerical results.

1.2. Outline

This article is organized as follows. In Section 2, we introduce the notations used in this work then we give the construction of the finite volume schemes we consider. We give the general form of conservative finite volume schemes that lead to a symmetric non-linear system (and thus derive from the minimization of a discrete functional approaching J), and that are consistent with piecewise affine functions. The symmetry property is crucial in order to ensure that for any given data, the scheme has a unique solution (Th. 2.12).

The construction of the schemes is summed up in Definition 2.6 or equivalently in (31) which gives the explicit formula for the discrete energy associated to the scheme.

Section 3 is concerned with the proof of error estimates for the approximate solutions of problem (1) in the case whereu∈W^2,p(Ω). This assumption is a natural extension of the usual H²(Ω) regularity for the Laplace equation (casep= 2), which is studied for example in [10]. Furthermore, it is proved in [4] that, when 1< p≤2 andf ∈L^q(Ω) (q >2) thenuactually belongs toW^2,p(Ω).

In this case we use similar methods than in [10] to prove error estimates, and we obtain the same kind of results than in the work by Glowinski and Marrocco [13] in the finite elements framework, namely a convergence order inh^p−11 whenp≥2 (Th. 3.1).

Notice that in the finite elements case, the convergence orders given in [13] have been improved by Barrett and Liu in [4, 5] and by Chow in [6]. The method used by S. Chow cannot be used in the finite volume case to improve Theorem 3.1 but it is useful to obtain error estimates for less regular solution. Indeed, in [2], we use it in the framework of Besov spaces, which are the natural regularity spaces for the p-Laplacian (see [16]).

We conclude this paper by giving some numerical results in Section 4.

1.3. Remarks

Of course, many physical applications occur in domains with more complex geometry and the use of rect- angular meshes seems to be too restrictive. Nevertheless, it is well known (especially in fluid mechanics) that one can cope with complex geometries with Cartesian meshes using a penalization method. More precisely, in order to solve (1) in any bounded domain Ω, we takeU a rectangle containing Ω and we solve instead of (1) the penalized problem 





−div

|∇uε|^p−2∇uε + 1

ε^p1_U\Ω|uε|^p−2u_ε=f, inU, u_ε= 0, on∂U.

(4) Note that the penalization term is monotone so that this problem has a unique solution. Furthermore one can easily show that

∇u_εL^p_(U\Ω)+1

εu_εL^p_(U\Ω)+∇u− ∇u_εL^p_(Ω)≤Cε^1p. (5) Hence, one can use our schemes on a rectangular mesh onU to solve (4) which gives a good approximation of the solution to (1). Notice that the schemes we propose in this paper can be easily extended to the problem (4) without changing their basic properties. In particular, all the error estimates remain valid for the penalized problem. Notice that the estimate (5) may not be optimal. Indeed, for the Laplace equation as for the Stokes problem, we can obtain sharper results by studying the corresponding boundary layer problem (see for instance [1]). We do not address this particular problem in this paper.

This method can be useful in many situations. Indeed, the practical computation of the approximate solution on Cartesian meshes can be easily optimized using multigrids methods and/or parallel computing. Furthermore, some glaciology models consist in free boundary problems in which a stationary non-linear diffusion equation like (1) has to be solved at each time step in a time dependent domain (see for instance [15]). Such problems can be handled with our methods by computing the solution to (4) on a fixed Cartesian mesh with a time dependent penalization term. Finally, the extension of our schemes to the 3D case would be much more simple than for unstructured grids.

We also want to emphasize the fact that our study can be applied to more general non-linear diffusion equation of the form

−div(k(|∇u|)∇u) =f,

provided thatk:R→Ris smooth enough and satisfies some monotony properties (see [14]).

2. Construction of the finite volume schemes

Let Ω be a rectangular bounded domain of R². We consider T a set of disjoint rectangular control volumes

K⊂Ω such that∪K¯= Ω. We denote byhthe maximum of the diameters of the control volumes inT.

LetxK be the center of the control volumeK. For all adjacent control volumesKand L, denote byK|Lthe edge betweenKandL.

In order to take the boundary conditions into account, we introduce artificial points constructed by symmetry with respect to the boundaries of Ω, as shown in Figure 1. This procedure is useful to state the scheme in a uniform way in the whole domain. We will give the details below.

The dual mesh T^∗ of T is defined to be the set of rectangle dual control volumes whose vertices are the pointsx_K as well as the artificial points introduced above, see Figure 1.

00 11 00

11 00

11 01 0000

1111 00

11 00

11 00 11 0000 1111 0000 1111

0

1 01 0

1 00 11

00 11

00 11

00 11 00

11

0000 1111 0000 1111 00 11 0

1 0011 0

1 0

1 0

1

0000 1111 00

11

0000 1111 00

11

0000 1111 0000 1111 00 11

0000 1111 00

11 01

00000000 00000000 0000 11111111 11111111 1111

00000000 11111111 000000 111111

0 1

0 1 0

1

0 1 0 1 00

11 00 11 0 1 K∗

K

x_K

u₁ −u1

u₁ −u₁

u₁

−u₁

−u4

u₄

Figure 1. Non uniform rectangular mesh and treatment of the boundary conditions.

00 11

0 1 00 11

0 1 00

0 11 1

0 1 K

VK

σ_K^v_,K∗

νK

K∗

K

σ^h_K,K∗

xK

ν_K

meshT

dual meshT^∗

Figure 2. Notations.

Furthermore, for any control volumeK∈ T, we define (see Fig. 2):

• m(K) the measure ofK;

• νKthe outward unit normal vector on∂K;

• VK={K∗∈ T^∗/m(K∩K∗)= 0}, the set of the four dual control volumes around the control volumeK;

• For any K∗ ∈VK,σ^h_K,K∗ (resp. σ^v_K,K∗) is the horizontal (resp. vertical) half-edge of ∂Kincluded in K∗

andEK={σ^h_K,K∗, σ_K^v_,K∗ /K∗∈VK}is the set of all the half-edges included in∂K.

0000 1111

0000 1111 0000 1111 0000

1111

l^K∗₃

l^K∗₄ l^K∗₂

σ₂^K∗

σ₁^K∗

x^K∗₂

x^K∗₄ x^K∗₃

K4,K∗ K3,K∗

K1,K∗ K2,K∗

σ^K∗₄ x^K∗₁ σ^K∗₃

l^K∗₁ =l^K∗₅

Figure 3. Notations in a dual control volumeK∗.

We will see in the sequel that the schemes we propose are naturally written in each dual control volume. That is the reason why, given a dual control volumeK∗, we introduce some notations to describe the situation inK∗

(see Fig. 3):

• (x^K∗_i )_i=1,2,3,4 are the vertices of the dual control volumeK∗ numbered counter clockwise starting from the lower left hand corner;

• (K^K∗i )_i=1,2,3,4are the corresponding control volumes with centers (x^K∗_i )_i=1,2,3,4;

• l^K∗_i is the distance betweenx^K∗_i andx^K∗_i+1;

• σ^K∗_i is the half-edge betweenK^K∗i andK^K∗i+1 located in K∗;

• m^K∗_i is the measure ofσ^K∗_i .

Conventionally, in a given dual control volume, the indicesi∈Zare understood modulo 4.

The finite volume method associates to all control volumes K an unknown value u_K. We denote by u^T = (u_K)_K∈T ∈R^T the approximate solution on the meshT.

For any continuous function v on Ω, we will also denote byv^T = (vK)_K∈T, withvK =v(xK), the projection ofv on the spaceR^T of discrete functions.

• Boundary conditions:

We only consider in this paper the homogeneous Dirichlet condition. For a given discrete functionu^T ∈R^T, which is a set of values on each control volumeK∈ T, the boundary conditions are taken into account by using the so-called ghost cells method. More precisely, we extend the values ofu^T on artificial points outside of Ω, as shown in Figure 1, each time it is needed.

Given a dual control volumeK∗, we define the projection operatorT_K∗ which associates to eachu^T ∈R^T its values in the four control volumes (K^K∗i )_i aroundK∗

T_K∗(u^T)^def=

u^T_1,K∗, u^T_2,K∗, u^T_3,K∗, u^T_4,K∗

. (6)

Notice that for boundary dual control volumes, we need the ghost cells described above in order to give sense to the definition of T_K∗. For instance, ifK∗ is located at the right boundary of Ω, then

u^T_2,K∗ =−u^T_1,K∗, u^T_3,K∗ =−u^T_4,K∗, as shown in Figure 1.

• Regularity of mesh families:

In the sequel, we will prove error estimates for families of meshes which satisfy the following regularity assumption:

There exists a constantc₁ (independent of the meshT) such that

m(K)≥c₁h², ∀K∈ T. (7)

Remark 2.1. Under the previous regularity assumption, there exist constants c₂, c₃ independent of T such that, for all dual control volumeK∗∈ T^∗,

diam(K∗)≤c₂handm(K∗)≥c₃h². 2.2. Discrete norms and classical inequalities

Each discrete function u^T ∈ R^T can be considered as a (piecewise constant) function in L^r(Ω) (for any r∈[1,+∞]) by noting abusively

u^T(z) =

K∈T

u_K1_K(z), ∀z∈Ω.

Then, we naturally define

u^T_L^r =

K∈T

m(K)|uK|^{r 1}_r

, ∀r <+∞, and

u^TL^∞ = sup

K∈T|u_K|.

Now let us define a discrete Sobolev norm for the elements ofR^T. For anyu^T ∈R^T, and anyK∗∈ T^∗, denote byδ^K∗_i (u^T) the differential quotient

δ_i^K∗(u^T) =u^T_i+1,K∗−u^T_i,K∗

l^K∗_i , ∀i∈ {1, . . . ,4}.

One can remark thatδ^K∗₁ and −δ₃^K∗ are approximations of the horizontal derivative operator∂_x whileδ₂^K∗ and

−δ^K∗₄ are approximations of the vertical derivative operator∂_y.

Definition 2.2. For anyu^T ∈R^T and anyK∗∈ T^∗, we introduce an approximation of|∇u| onK∗ defined by

|u^T|_1,K∗= 1 2

4 i=1

δ^K∗_i (u^T)^{2 1}₂

.

The discreteW₀^1,pnorm ofu^T is then defined by u^T_1,p,T =

K∗∈T^∗

m(K∗∩Ω)|u^T|^p_1,K∗

_p¹ .

Remark 2.3.

• The Dirichlet boundary conditions are taken into account in the definition of the discrete W₀^1,p norm through the introduction of the ghost cells. Actually, this implies that · _1,p,T is a norm on R^T.

• For any function v∈W₀^1,p(Ω), withp >2, we have

v^T_1,p,T ≤Cv_W^1,p, (8)

wherev^T = (v(x_K))_K∈T is the projection ofvon the meshT, andCdepends only on the mesh regularity constant c₁ in (7).

Let us state the discrete version of the Poincar´e inequality. This result is classical in the case p= 2 (see for example [10]). Whenp= 2, it is proved in a slightly different context in [3]. Note that one does not assume any regularity assumptions on the mesh.

Lemma 2.4 (discrete Poincar´e inequality). Let T be a mesh of the rectangle Ω. There exists a constant C which only depends on psuch that for anyu^T ∈R^T, we have

u^TL^p≤Cdiam(Ω)u^T1,p,T.

Proof. It is a straightforward adaptation from the proof in [3], taking into account the ghost cells method used

for the boundary conditions.

2.3. Construction of the schemes

A family of nine-points finite volume schemes is proposed to compute approximate solutions of problem (1).

Such schemes are obtained by integrating equation (1) on each control volumeK∈ T:

K

f(z) dz=

K

−div

|∇u|^p−2∇u dz=−

∂K

|∇u|^p−2∇u·ν_Kds

=

K∗∈VK

−

σ^h_K,K∗|∇u|^p−2∇u·ν_Kds−

σ_K,K∗^v|∇u|^p−2∇u·ν_Kds

. (9)

Then, the finite volume method consists in approaching the exact relation between the fluxes (9) by a system of discrete equations

aK(u^T)^def=

K∗∈VK

aK,^K∗(u^T) =m(K)fK, ∀K∈ T, (10) whereaK,^K∗(u^T) is a numerical flux to be determined which is supposed to approach the exact flux, that is:

a_K,K∗(u^T)≈ −

σ_K,K∗^h|∇u|^p−2∇u·ν_Kds−

σ^v_K,K∗|∇u|^p−2∇u·ν_Kds, (11)

andfKdenotes

f_K= 1 m(K)

K

f(z) dz. (12)

2.3.1. General structure

The classical finite volume theory for the Laplace equation (see [11] for instance) provides very natural ways to approach the normal derivative ∇u·νK on the edges. On the other hand, in the literature there is no systematic method to discretize the norm of the gradient of the solution.

In this paragraph, we describe a wide class of possible choices for the numerical fluxesa_K,K∗(u^T) in which the approximations of the terms∇u·ν_Kand|∇u|^p−2are chosen independently. Actually, in the next paragraph we

show that these two discretizations have to be compatible (in a sense to be precised) to ensure good properties of the schemes, in particular the well-posedness of the discrete non-linear equations.

The general framework we propose is the following.

(1) On each dual control volumeK∗, we choose to approximate|∇u|² by a quadratic form (to be precised later) in the variables (u^T_i,K∗){i=1,2,3,4}, namely

|∇u|²≈qK∗(TK∗(u^T)) onK∗, (13)

whereT_K∗(u^T) is defined at the end of Section 2.1. This leads to a first approximation of the fluxes on any half edgeσ⊂K∗ by

−

σ|∇u|^p−2∇u·νKds≈ −qK∗(TK∗(u^T))^p−22

σ∇u·νKds.

(2) The problem is now reduced to the approximation of the same linear fluxes as for the Laplace equation.

We suppose that we are given linear formsA^h_K,K∗,A^v_K,K∗, in the variables (u^T_i,K∗){i=1,2,3,4}, such that

−

σ_K,K∗^h∇u·νKds≈A^h_K,K∗(TK∗(u^T)) and −

σ^v_K,K∗∇u·νKds≈A^v_K,K∗(TK∗(u^T)). (14) (3) The numerical flux satisfying (11) is then given by

a_K,K∗(u^T)^def=q_K∗(T_K∗(u^T))^p−22

A^h_K,K∗(T_K∗(u^T)) +A^v_K,K∗(T_K∗(u^T)) .

2.3.2. Choice of qK∗ andA^h_K,K∗,A^v_K,K∗

First of all, it is convenient to representqK∗ through a symmetric non negative matrix onR⁴denoted byBK∗

so that we have

q_K∗(v) = (B_K∗v, v), ∀v∈R⁴.

Notice that the schemes proposed above can also be written as a sum of contributions on each dual control volume. Indeed, each numerical fluxa_K,K∗(u^T) depends only on the valuesT_K∗(u^T) = (u^T_i,K∗){i=1,2,3,4}ofu^T at the four vertices of the dual control volumeK∗. More precisely, one can easily see that the mapa:u^T ∈R^T → (a_K(u^T))_K∈R^T defined in (10) which represents the scheme reads

a(u^T) =

K^∗∈T^∗

m(K∗∩Ω)T_K∗^t ◦aK∗◦TK∗(u^T) (15) with

aK∗(v)^def= 1

m(K∗)(BK∗v, v)^p−22







A^h_K1,K∗v+A^v_K1,K∗v A^h_K2,K∗v+A^v_K2,K∗v A^h_K3,K∗v+A^v_K3,K∗v A^h_K4,K∗v+A^v_K4,K∗v





= 1

m(K∗)(BK∗v, v)^p−22 AK∗v, ∀v∈R⁴, (16) whereA_K∗ is the corresponding 4×4 matrix.

We first derive some properties of BK∗ and AK∗, which ensure the consistency of the scheme but also the existence and uniqueness of a solution of the scheme.

• Consistency of qK∗:

By definition,BK∗ is a symmetric non-negative matrix. Moreover,qK∗ is supposed to approach|∇u|²so that is natural to impose the following consistency property:

Approximation (13) is exact for affine functions onK∗, andq_K∗(v) = 0 if and only ifv₁=v₂=v₃=v₄. (17)

In particular, this imposes that the kernel ofB_K∗ is spanned by the vector e₀ = (1,1,1,1)^t, so that the rank ofB_K∗ is 3.

• Conservativity of the linear fluxes:

Any reasonable finite volume scheme has to be locally conservative, in our case it means that, for any dual control volumeK∗, the linear fluxes have to be in the following form









A^h_K1,K∗ = A_4,K∗, A^v_K1,K∗=−A1,K∗, A^h_K2,K∗ =−A_2,K∗, A^v_K2,K∗= A_1,K∗, A^h_K3,K∗ = A_2,K∗, A^v_K3,K∗=−A_3,K∗, A^h_K4,K∗ =−A_4,K∗, A^v_K4,K∗= A_3,K∗,

(18)

where A_i,K∗ is an approximation of the linear flux

σ^K∗_i∇u·ν_i^K∗ds, ν_i^K∗ being the unit normal to σ^K∗_i oriented conventionally fromK^K∗i toK^K∗i+1 (see Fig. 3).

• Construction of the linear fluxes:

To ensure the consistency of the numerical fluxesA_i,K∗, the approximation has to be exact on affine functions.

One easily checks that this requirement amounts to A_i,K∗ =m^K∗_i µ^K∗_i u^T_i+1,K∗ −u^T_i,K∗

l^K∗_i +

1−µ^K∗_i

u^T_i+2,K∗−u^T_i+3,K∗

l_i+2^K∗

, (19)

where µ^K∗_i is a real parameter. Notice that the classical finite volume approximation of the Laplace equation (i.e. whenp= 2) corresponds to the caseµ^K∗_i = 1,∀i∈ {1, . . . ,4},∀K∗∈ T^∗.

For anyK∗∈ T^∗ and anyi, let us define

τ_i^K∗= m^K∗_i

l^K∗_i , (20)

which depends only on the geometrical structure of the mesh. Using (18) the matrixAK∗ defined in (16) reads for each dual control volumeK∗







τ₁µ₁+τ₄µ₄ −τ₁µ₁+τ₄(1−µ₄) −τ₁(1−µ₁)−τ₄(1−µ₄) −τ₄µ₄+τ₁(1−µ₁) τ₂(1−µ₂)−τ₁µ₁ τ₂µ₂+τ₁µ₁ −τ₂µ₂+τ₁(1−µ₁) −τ₂(1−µ₂)−τ₁(1−µ₁)

−τ3(1−µ3)−τ₂(1−µ2) τ₃(1−µ3)−τ₂µ₂ τ₃µ₃+τ₂µ₂ −τ3µ₃+τ₂(1−µ2) τ₃(1−µ3)−τ₄µ₄ −τ4(1−µ4)−τ₃(1−µ3) τ₄(1−µ4)−τ₃µ₃ τ₄µ₄+τ₃µ₃





 (21)

where we dropped the subscriptK∗ forτ_i^K∗ andµ^K∗_i .

• Symmetry property of the scheme:

One of the fundamental properties of the continuous equation (1) is that it derives from a minimization problem. It is natural to look for numerical schemes which preserve this property.

The scheme (28) is the Euler-Lagrange equation for critical points of a functionalJ_T :R^T →Rif and only if the differential of the map

a:u^T = (uK)K∈T →(aK(u^T))K∈T,

is symmetric at each point. In that case, we shall see later that we can give an explicit formula for J_T, and that J_T is strictly convex and coercive.

For non-symmetric schemes, the existence and uniqueness of a solution to the non-linear discrete equations is not guaranteed. Even if the solution exists, its practical computation is not obvious. That is the reason why, from now on, we focus our attention on symmetric schemes.

We now state the following proposition which characterizes such schemes in our framework. The proof is given in Section 2.6.

Proposition 2.5. The differential of the mapa:u^T = (uK)K∈T →(aK(u^T))K∈T is symmetric at each point if and only if one of the two following conditions is fulfilled:

• p= 2and AK∗ is a symmetric matrix for anyK∗∈ T^∗;

• p= 2and for any K∗∈ T^∗ there exists aλK∗ ∈Rsuch that A_K∗ =λ_K∗B_K∗.

In order for the scheme to be consistent with the balance equation (9) we will see later that, necessarily, λK∗ =m(K∗).

Hence, at this point, the choice of the discretization of the linear part of the fluxes across half-edges in each dual control volume (i.e. the choice of the parametersµ^K∗_i in (19)) determines the scheme completely. In other words, if we want the scheme to be symmetric we cannot choose independently the discretization of the norm of the gradient|∇u|(represented by B_K∗) and the discretization of the normal derivative∇u·ν_K (represented byA_K∗).

Furthermore, Proposition 2.5 implies that A_K∗ is necessarily symmetric. The expression (21) shows that the (µ^K∗_i )_i have to fulfill the following constraints

τ₄^K∗(µ^K∗₄ −1) =τ₂^K∗(µ^K∗₂ −1),

τ₃^K∗(µ^K∗₃ −1) =τ₁^K∗(µ^K∗₁ −1). (22) The scheme on the dual control volume K∗ now seems to be determined by the choice of µ^K∗₁ and µ^K∗₄ for instance. Nevertheless, using (22), it is easy to see that the matrix AK∗ given by (21) depends only on the quantityτ₁^K∗µ^K∗₁ +τ₄^K∗µ^K∗₄ . Consequently, without loss of generality the coefficients (µ^K∗_i )_ican be chosen so that τ₁^K∗(µ^K∗₁ −1) =τ₂^K∗(µ^K∗₂ −1) =τ₃^K∗(µ^K∗₃ −1) =τ₄^K∗(µ^K∗₄ −1)^def=ξ^K∗. (23) Therefore, for anyK∗ the matrix AK∗ only depends on a real parameterξ^K∗ in the following way:

A_K∗ =







2ξ^K∗+τ₁^K∗+τ₄^K∗ −2ξ^K∗−τ₁^K∗ 2ξ^K∗ −2ξ^K∗−τ₄^K∗

−2ξ^K∗−τ₁^K∗ 2ξ^K∗+τ₁^K∗+τ₂^K∗ −2ξ^K∗−τ₂^K∗ 2ξ^K∗

2ξ^K∗ −2ξ^K∗−τ₂^K∗ 2ξ^K∗+τ₂^K∗+τ₃^K∗ −2ξ^K∗−τ₃^K∗

−2ξ^K∗−τ₄^K∗ 2ξ^K∗ −2ξ^K∗−τ₃^K∗ 2ξ^K∗+τ₄^K∗+τ₃^K∗





, (24)

andB_K∗ is given by Proposition 2.5.

• Consistency of the scheme:

As we have claimed above, the consistency of the whole scheme determines the value of the coefficientλ_K∗

introduced in Proposition 2.5.

Indeed, letwbe an affine function in a dual control volumeK∗, whose gradient is (α, β)^t, and letv=T_K∗w^T. We easily check that

v₂=v₁+αl^K∗₁ , v₃=v₁+αl^K∗₁ +βl^K∗₂ , v₄=v₁+βl₂^K∗, so that, using (24), we have

A_K∗v=







−m^K∗₄ β−m^K∗₁ α

−m^K∗₂ β+m^K∗₁ α m^K∗₂ β+m^K∗₃ α m^K∗₄ β−m^K∗₃ α





.

Asl^K∗₁ =m^K∗₂ +m^K∗₄ andl^K∗₂ =m^K∗₁ +m^K∗₃ (see Fig. 3), we get

(A_K∗v, v) =−m^K∗₁ α(v₁−v₂)−m^K∗₃ α(v₄−v₃)−m^K∗₂ β(v₂−v₃)−m^K∗₄ β(v₁−v₄)

=l₁^K∗m^K∗₁ α²+l^K∗₁ m^K∗₃ α²+m^K∗₂ l^K∗₂ β²+m^K∗₄ l^K∗₂ β²

=l₁^K∗l^K∗₂ (α²+β²) =m(K∗)|∇w|². By Proposition 2.5, we also haveAK∗ =λK∗BK∗ so that

λ_K∗(B_K∗v, v) =m(K∗)|∇w|². Hence, the consistency requirement (17) holds if and only ifλK∗ =m(K∗).

• Admissible schemes:

Given a dual control volume K∗ and a value of ξ^K∗, the matrices A_K∗ and B_K∗ are defined by (24) and by B_K∗ =_m(¹_K∗)A_K∗. It remains to check thatB_K∗ is a non negative matrix of rank 3, as required in (17).

It is easy to check that the kernel of the matrix BK∗ defined above contains e₀. Consider now the qua- dratic form v → (BK∗v, v) restricted to the hyperplane {e0}^⊥ orthogonal to the vector e₀. In the basis (−1,1,1,−1)^t,(1,1,−1,−1)^t,(1,−1,1,−1)^t, its matrix reads

4 m(K∗)





τ₁^K∗+τ₃^K∗ 0 τ₃^K∗ −τ₁^K∗

0 τ₂^K∗ +τ₄^K∗ τ₄^K∗ −τ₂^K∗

τ₃^K∗−τ₁^K∗ τ₄^K∗ −τ₂^K∗ 8ξ^K∗+τ₁^K∗+τ₂^K∗+τ₃^K∗+τ₄^K∗



. (25)

The computation of main minor determinants show that this matrix is positive definite if and only if 2ξ^K∗+m^K∗₁ m^K∗₃ +m^K∗₂ m^K∗₄

l^K∗₁ l₂^K∗ >0.

This gives a condition onξ^K∗ which ensures the admissibility of the scheme. It will be shown that this condition guarantees existence and uniqueness for the approximate solutionu^T as well as the discreteW^1,pestimate (see Lems. 2.8 and 3.3).

2.3.3. Admissible finite volume schemes

Let us sum up our construction of admissible schemes in the following definition. We recall that the projec- torT_K∗ is defined in (6). In particular, the boundary conditions are taken into account in the scheme through the definition of this projector for boundary dual control volumes.

Definition 2.6. For each dual control volumeK∗, we suppose given a real numberξ^K∗ such that 2ξ^K∗+m^K∗₁ m^K∗₃ +m^K∗₂ m^K∗₄

l^K∗₁ l₂^K∗ >0. (26)

Recall that τ_i^K∗=^m_lK∗^K∗i

i and consider the matrixAK∗ defined by (24) andBK∗ by B_K∗ = 1

m(K∗)A_K∗. (27)

The finite volume scheme associated to the parameters (ξ^K∗)_K∗∈T∗ is then defined by

a(u^T) = (m(K)f_K)_K∈T , (28)

wherea=

K^∗∈T^∗

m(K∗∩Ω)T_K∗^t ◦a_K∗◦T_K∗,and

aK∗(v) = (BK∗v, v)^p−22 BK∗v, for allv∈R⁴.

Note that the admissibility condition (26) is fundamental to give sense to the definition ofa_K∗ since it ensures that (B_K∗v, v)≥0 for anyv∈R⁴.

2.4. Discrete energy

2.4.1. Construction of the discrete energy

The schemes defined in Definition 2.6 are built in such a way that the symmetry of the differential of the mapaonR^T is ensured. Hence, there exists a discrete functionalJ_T :u^T ∈R^T →R, that we call the discrete energy of the system, such that we have

a_K(u^T)−m(K)f_K=∂J_T

∂u_K, ∀K∈ T. (29)

Using the homogeneity of the map a, it is possible to obtain an explicit formula for J_T. Indeed, let us check that the functional defined by

J_T(u^T) =1 p

K∈T

a_K(u^T)u_K−

K∈T

m(K)f_Ku_K

is the discrete energy of the scheme (which is unique up to a constant). An easy computation shows that

∇J_T(u^T) =1

p(da(u^T))^t.u^T +1

pa(u^T)−(m(K)f_K)_K. (30) Sincea: (u_K)_K→(a_K(u^T))_Kis a positively homogeneous function of degree p−1, we have

da(u^T).u^T = (p−1)a(u^T).

By construction (see Prop. 2.5), we know thatda(u^T) is symmetric so that, we obtain (da(u^T))^t.u^T =da(u^T).u^T = (p−1)a(u^T).

Hence, (30) yields (29).

Let us now give a more precise expression for the energyJ_T. Lemma 2.7. Let u^T, v^T ∈R^T, we have

a(u^T)·v^{T def}=

K∈T^∗

a_K(u^T)v_K=

K∗∈T^∗

m(K∗∩Ω)(B_K∗T_K∗(u^T), T_K∗(u^T))^p−22 (B_K∗T_K∗(u^T), T_K∗(v^T)).

Proof. It is a straightforward application of formulas (15) and (27).

Denote byB

1 2

K∗ the square root of the non-negative symmetric matrixBK∗. From Lemma 2.7, we deduce that J_T(u^T) = 1

p

K∗∈T^∗

m(K∗∩Ω)|B_K∗¹² T_K∗(u^T)|^p−

K∈T

m(K)f_Ku_K. (31)

It is now clear that this discrete functional can be seen as an approximation of the continuous functional J associated to problem (1).