Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids

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Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids

Robert Eymard, Thierry Gallouët, Raphaele Herbin

To cite this version:

Robert Eymard, Thierry Gallouët, Raphaele Herbin. Cell centred discretisation of non linear ellip- tic problems on general multidimensional polyhedral grids. Journal of Numerical Mathematics, De Gruyter, 2009, 17 (3), pp.173–193. �10.1515/JNUM.2009.010�. �hal-00351867�

Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids

R. Eymard^†, T. Gallou¨et^‡and R. Herbin^§ January 10, 2009

Abstract

This work is devoted to the discretisation of non linear elliptic problems on general polyhedral meshes in several space dimensions. The SUSHI scheme which was recently studied for anisotropic hetero- geneous problems is applied in its full barycentric version, thus resulting into a cell centred scheme written under variational form, also known as ’SUCCES’. We prove the existence of the approximate solution and its convergence to the weak solution of the continuous solution as the mesh size tends to 0. Numerical examples are shown for thep-Laplacian.

1 Introduction

We study the following problem: find an approximation ofu, weak solution to the following equation:

−diva(x,∇u) =f in Ω,

u= 0 on ∂Ω, (1)

where we denote by ∂Ω = Ω\Ω the boundary of Ω, under the following assumptions:

Ω is an open bounded connected polygonal subset of R^d, d∈N^⋆, (2a)

a : Ω×R^d→R^d is a Caratheodory function, (2b)

(e.g. a function such that for a.e. x∈Ω, ξ 7→a(x, ξ) is continuous, and for anyξ ∈R^d the function x7→a(x, ξ) is measurable)

∃a∈(0,+∞), p∈(1,+∞) ; a(x,ξ)·ξ≥a|ξ|^p, for a.e. x∈Ω, ∀ξ ∈R^d, (2c) (a(x,ξ)−a(x,χ))·(ξ−χ)>0, for a.e. x∈Ω, ∀ξ,χ∈R^dwith ξ6=χ, (2d)

∃a∈L^p′(Ω), Λ∈(0,+∞) ; |a(x,ξ)| ≤a(x) + Λ|ξ|^p−1, for a.e. x∈Ω, ∀ξ∈R^d, (2e) and

f ∈L^p′(Ω) where p^′ = p

p−1. (2f)

∗This work was supported by Groupement MOMAS, CNRS/PACEN

†Universit´e Paris Est, France, [email protected]

‡Universit´e de Provence, France, [email protected]

§Universit´e de Provence, France, [email protected]

If the function asatisfies (2b)-(2e), then the mapping u7→ −diva(·,∇u) is a Leray-Lions operator, a classical example of which is thep-Laplacian operator, obtained by setting

a(x, ξ) =|ξ|^p−2ξ, ∀ξ ∈R^d\ {0}, for a.e. x∈Ω,

a(x,0) = 0, for a.e. x∈Ω. (3)

It is well-known [27] that under hypotheses (2), there exists a unique weak solution (1), that is a functionu satisfying:





u∈W₀^1,p(Ω), Z

Ω

a(x,∇u(x))· ∇v(x)dx= Z

Ω

f(x)v(x)dx, ∀v∈W₀^1,p(Ω). (4) The numerical approximation of (4) has been the subject of numerous works: finite element methods have been extensively studied [23, 24, 22, 11, 28, 9, 25, 16]. Finite volume schemes have also been addressed, first on Cartesian grids [5, 6]; more recent works [10, 8, 7] are concerned with the DDFV (Discrete Duality Finite Volume) scheme for general two-dimensional grids, introduced in [26] and first analysed in [12] for linear problems; these schemes require the use of two dual grids and sets of unknowns. In [13] the mixed finite volume scheme involving unknowns at the faces of the mesh, first introduced and analysed in [14] is shown to converge for general multidimensional grids.

In this paper, we study a numerical scheme which satisfies the following properties:

• It is a cell centred scheme, that is there is only one unknown associated to each grid cell. The advantage of such method is that it can be more easily extended to nonlinear coupled problems.

• It is defined for general, possibly non-conforming, polygonal meshes in any space dimension.

• The approximate problem has one and only one solution, so that the resulting nonlinear system of equations may be solved by an adequate (iterative) method.

• The convergence of the approximate solution and gradient to the exact solution and gradient when the mesh size goes to 0 are proven.

The scheme ”SUCCES” (Scheme Using Conservativity and Consistency error Stabilisation) that we use here was first introduced in [21] for the discretisation of the Laplace operator in the incompress- ible Navier-Stokes equations on general multidimensional grids. Its numerical performance for the numerical simulation of flow in heterogeneous porous media is demonstrated in [3]. Its convergence analysis for linear problems is carried out in [19] in the more general framework of the SUSHI (Scheme Using Stabilisation and Hybrid Interfaces) scheme, for which edge unknowns may be present or not, according to the user’s choice.

The contents of this paper is the following. In Section 2, we present the discretisation scheme and a few basic properties. Some estimates (using discrete Sobolev embeddings) are presented in Section 3, along with the proof of the existence and uniqueness of the approximate solution. In Section 4, we prove the convergence results, which are based on some “discrete functional analysis” tools, mimicking classical functional analysis tools: the Kolmogorov theorem is used to get some compactness results (strong convergence for the approximate solution, weak convergence for the approximate gradient).

Then the well-known ’Minty trick’ [27] is used to overcome the problem of passing to the limit in a nonlinear monotone problem under only weak convergence properties for the approximate gradient of the solution. Finally, the Leray-Lions trick of [27] allows us to recover the strong convergence of this approximate gradient. The numerical behaviour of the scheme is shown on an example in Section 5.

2 The finite volume scheme

Let us first begin with the detailed description of the mesh under consideration in this paper.

Definition 2.1 (Space discretisation) Let Ωbe a polyhedral open bounded connected subset of R^d, withd∈N\ {0}, and∂Ω = Ω\Ωits boundary. A discretisation of Ω, denoted by D, is defined as the triplet D= (M,E,P), where:

1. Mis a finite family of non empty connected open disjoint subsets of Ω (the “control volumes”) such that Ω = ∪_K∈MK. For any K ∈ M, let ∂K =K\K be the boundary of K; let |K|>0 denote the measure of K and h_K denote the diameter of K.

2. E is a finite family of disjoint subsets of Ω (the “edges” of the mesh), such that, for allσ ∈ E, σ is a non empty open subset of a hyperplane of R^d, whose (d-1)-dimensional measure |σ| is strictly positive. We also assume that, for all K ∈ M, there exists a subset E_K of E such that

∂K =∪_σ∈EKσ. For any σ ∈ E, we denote by M_σ ={K ∈ M, σ∈ E_K}. We then assume that, for all σ ∈ E, either M_σ has exactly one element and then σ⊂∂Ω (the set of these interfaces, called boundary interfaces, is denoted by Eext) orMσ has exactly two elements (the set of these interfaces, called interior interfaces, is denoted by E_int). For all σ ∈ E, we denote by x_σ the barycentre of σ. For all K ∈ M and σ ∈ E_K, we denote by n_K,σ the unit vector normal to σ outward to K.

3. P is a family of points ofΩindexed by M, denoted byP = (x_K)_K∈M, such that for allK ∈ M, x_K ∈K and K is assumed to be x_K-star-shaped, which means that for all x∈K, the property [x_K,x] ⊂ K holds. Denoting by d_K,σ the Euclidean distance between x_K and the hyperplane including σ, one assumes thatd_K,σ >0. We then denote by D_K,σ the cone with vertex x_K and basis σ.

Remark 2.1 (Non convex generalized hexahedra) The above definition applies to a large va- riety of meshes. Note that no hypothesis is made on the convexity of the control volumes; in fact, generalised hexahedra, i.e. with faces which may be composed of several planar sub-faces may be used. Often encountered in underground flow simulations, such hexahedra may have up to 12 faces (resp. 24 faces) if each non planar face is composed of two triangles (resp. four triangles), but only 6 neighbouring control volumes.

LetD= (M,E,P) be a discretisation of Ω in the sense of Definition 2.1. The size of the discretisation D is defined by:

h_D = sup{h_K, K ∈ M}.

Let HD ⊂L²(Ω) be the set of piece-wise constant functions on the control volumes of the mesh and let u_K denote the (constant) value ofu on K.

A first simple idea to find a scheme that will approximate (1) is to use the usual non conforming Galerkin method: assuming that forv∈H_D we know how to construct an adequate discrete gradient

∇_Dv (expected to be an approximation of the gradient of the exact solution), we then seeku_D ∈H_D

such that Z

Ω

a(x,∇_Du_D(x))· ∇_Dv(x) dx= Z

f(x)v(x) dx, ∀v∈H_D. (5) The construction of the discrete gradient is obtained from the values of the discrete function on the cell and from its reconstructed values on the edges. These are constructed in the following way. For K ∈ M, one chooses x_K ∈ M (this choice is possible for any set K which is “x_K star shaped”). For any interior edge (interface) σ of M, choose some points x_M of the mesh close to σ and write x_σ (recall thatx_σ is the barycentre of σ) as a combination of these points: x_σ =P

M∈Mβ_σ^Mx_M. Note

that there is no need for this combination to be convex. The coefficients of the combination, however, should be bounded.

Then, for anyv∈H_D, one sets:

Π_σu=







 X

M∈M

β_σ^Mu_M ifσ ∈ E_int,

0 if σ∈ E_ext.

(6)

Next, denoting by Id thed×didentity matrix, we use the following geometrical relationship 1

|K|

X

σ∈EK

|σ|n_K,σ(x_σ−x_K)^t= Id,∀K ∈ M, (7) (which is an easy consequence of the Stokes formula), to define a consistent discrete gradient∇_Dv∈H_D of a functionv∈H_D as the piece-wise constant function equal to its constant value∇_Ku a.e. in K:

∇_Du(x) =∇_Kufor a.e. x∈K,∀K ∈ M. (8a)

∇_Ku= 1

|K|

X

σ∈EK

|σ|(Π_σu−u_K)n_K,σ. (8b)

We may then try to find u_D ∈ H_D solution of (5); however, it is easily seen that the problem lacks coercivity because of the definition of the discrete gradient. Take for instance d = 1, Ω = (0,1) and a uniform mesh with step size h = 1/N. For each cell Ki = ((i−1)h, ih) (i = 1, . . . , N), take x_Ki =x_i= (i−1)h+^h₂. A natural combination to compute the interface values isx_i+1/2 = ¹₂(x_i+x_i+1), i= 1, . . . , N−1 (and the boundary conditions give x1

2 =x_N+1

2 = 0). Then, the discrete gradient of v= (v_i)_i=1,N has the value _2h¹ (v_i+1−v_i−1) on cellK_i fori= 2, . . . , N−1, the value _2h¹ (v₁+v₂) on cell K₁ and the value ⁻¹_2h(v_N−1+v_N) on cellK_N. Taking v_i = (−1)ⁱ for alli leads to a discrete gradient equal 0 in all cells. This problem must be cured to obtain the coercivity of the discrete problem and the uniqueness of its solution.

A remedy to this problem was found by using a stabilisation which uses a consistency estimate [18, 21, 19]. Let us introduce, for any v∈H_D, for anyK∈ M and σ∈ E_K, the valueR_K,σv defined by:

R_K,σv= 1

d_K,σ(Πσv−v_K− ∇_Ku·(xσ−x_K)). (9) Using this value, we define the functionR_Dv by the constant valueR_K,σv in the cone D_K,σ:

RDv(x) =RK,σv for a.e. x∈DK,σ,∀K∈ M,∀σ ∈ EK. (10) We now consider the following approximate problem:

find u∈H_D such that hu, vi_D=

Z

Ω

a(x,∇_Du(x))· ∇_Dv(x) dx+b(u, v) = Z

f(x)v(x) dx, ∀v∈H_D. (11) with

b(u, v) = Z

Ω

|R_Du(x)|^p−1sgn(R_Du(x))R_Dv(x) dx, (12) introducing the function sgn(x) = 1 for allx∈]0,+∞[, sgn(x) =−1 for allx∈]−∞,0[ and sgn(0) = 0.

We now define, for all K∈ M, the function a_K : R^d→R^d

∀ξ ∈R^d,a_K(ξ) = _|K|¹ R

Ka(x, ξ) dx, (13)

we can then write that hu, vi_D= X

K∈M



|K|a_K(∇_Ku)· ∇_Kv+ X

σ∈EK

|σ|d_K,σ

d |R_K,σu|^p−1sgn(R_K,σu)R_K,σv



.

Remark 2.2 (An alternate scheme) We could also proceed as in [19]: define ∇_K,σv in the cone D_K,σ:

∇_K,σv=∇_Kv+R_K,σvn_K,σ,

then define∇_Dvby the constant value∇_K,σvin the coneD_K,σ, and then define the scheme by (5). This scheme presents similar properties of convergence to the one which is studied here, under conditions of regularity on the mesh which ensure that there exist two positive realsα, β independent of the mesh such that

αkvk_p,Π≤ k∇_Dvk_L^p_(Ω)≤βkvk_p,Π,

where k · k_p,Πis defined by (18). This can be shown under more restrictive hypotheses than those used here.

3 Estimates, existence and uniqueness of the approximate solution

We define the following discrete W₀^1,p norm on H_D: kuk^p_1,p,D= X

K∈M

X

σ∈EK

|σ|d_K,σ D_σu

d_σ p

, (14)

where d_σ = d_K,σ +d_L,σ, if M_σ = {K, L}, and d_σ = d_K,σ, if M_σ = {K} and D_σu = |u_K −u_L| if M_σ ={K, L}and D_σu=|u_K−0|ifM_σ ={K}.

Let us first recall the following results, which are proven in [19].

Lemma 3.1 (Discrete Sobolev inequality) Let d≥1, 1≤p <∞ and let Ω be a polyhedral open bounded connected subset of R^d. Let D be a mesh of Ω in the sense of Definition 2.1. Let η > 0 be such that η ≤ dK,σ/dL,σ ≤ 1/η for all σ ∈ E, where Mσ = {K, L}. Then, there exists q > p only depending onp (q= _d−p^pd , for instance, in the case 1< p < d, andq is any value in ]p,∞[in the case d≤p) and there existsC₁, only depending on d, Ω, p and η such that

kuk_L^q_(Ω) ≤C1kuk1,p,D ∀u∈HD, (15)

where kuk^p_1,p,D is defined by (14).

Lemma 3.2 (Compactness in L^p) Let d ≥ 1, 1 ≤ p < ∞ and Ω be a polyhedral open bounded connected subset of R^d. Let F be a family of meshes of Ω in the sense of Definition 2.1. Let η > 0 be such that, for all D ∈ F, one has η ≤ d_K,σ/d_L,σ ≤ 1/η for all σ ∈ E, where M_σ ={K, L}. For D ∈ F, let uD ∈HD and assume that there exists C∈R such, for all D ∈ F, kuDk1,p,D ≤C. Then, the family (u_D)_D∈F is relatively compact in L^p(Ω) and also in L^p(R^d) taking u_D = 0 outside Ω.

We introduce a measure of the regularity of the discretisation by θ_D = max

h_K d_K,σ,d_K,σ

d_L,σ, K ∈ M, σ∈ E_K∩ E_int,M_σ={K, L}

. (16)

Note that ifθ_D is bounded, the hypotheses of Lemmas 3.1 and 3.2 hold. We next define:

θ_D,Π= max θ_D, ( P

M∈M|β_σ^M||x_M −x_σ|²

h²_K , K ∈ M, σ∈ E_K∩ E_int )!

(17)

We then define the following discreteW₀^1,p norms onH_D, depending on the mappings Π_σ: kuk^p_p,Π= X

K∈M

X

σ∈EK

|σ|d_K,σ

|Π_σu−u_K| d_K,σ

p

(18) Let us remark that the following inequality holds:

kuk^p_1,p,D ≤(1 +θ_D)kuk^p_p,Π,∀u∈H_D, (19) using the following inequalities

|u_K−u_L| d_σ

p

≤ dK,σ

d_σ p^′

+ dL,σ

d_σ

p^′!p/p^′

|Πσu−u_K|^p

d^p_K,σ +|Πσu−u_L|^p d^p_L,σ

! ,

d_K,σ dσ

p^′

+ d_L,σ

dσ

p^′

≤ d_K,σ dσ

+d_L,σ dσ

= 1, and d_L,σ ≤θ_Dd_K,σ. We also have

k∇_Duk^p_Lp(Ω)≤d^p−1 kuk^p_p,Π,∀u∈H_D, (20) using the inequality

X

σ∈EK

|σ|d_K,σ

|K|

Π_σu−u_K d_K,σ n_K,σ

p

≤



X

σ∈EK

|σ|d_K,σ

|K| 1^p′





p/p^′

X

σ∈EK

|σ|d_K,σ

|K|

|Π_σu−u_K|^p

|d_K,σ|^p



.

Lemma 3.3 (Coerciveness) Under hypotheses (2), let D be a discretisation in the sense of Defini- tion 2.1. Let θ≥θD (defined by (16)). Then there exists C2 >0, only depending on d, Ω, p, aand θ such that

∀u∈H_D, C₂kuk^p_p,Π≤ hu, ui_D, (21) where hu, ui_D is defined in (11).

Proof. Thanks to (2c), we have Z

Ω

a(x,∇_Du(x))· ∇_Du(x) dx≥a X

K∈M

|K||∇_Ku|^p.

We get, from (12),

b(u, u) = X

K∈M

X

σ∈EK

|σ|d_K,σ

d |R_K,σu|^p. Using H¨older’s inequality, we have

∀x, y∈R,∀µ >0,|x+y| ≤ |x|+|y| ≤(1 +µ^p′)^1/p′(|x|^p+|y|^p µ^p )^1/p. We apply the above inequality to x=R_K,σu and y= _d¹

K,σ∇_Ku·(x_σ−x_K), for a valueµwhich will be chosen later. We get

|Π_σu−u_K| d_K,σ

p

≤(1 +µ^p′)^p/p′

|R_K,σu|^p+

|∇_Ku·(x_σ−x_K)|

µd_K,σ

p .

We then chooseµ= θ

a^1/p, which ensures |x_σ−x_K|

µdK,σ

p

≤a, ∀K∈ M,∀σ ∈ E_K.

We then get, for this value ofµ, (1 +µ^p′)^−p/p′ X

K∈M

X

σ∈EK

|σ|d_K,σ d

|Π_σu−u_K| d_K,σ

p

≤ hu, uiD,

which concludes the proof of (21).

Lemma 3.4 (Estimate, existence and uniqueness of the solution) Under hypotheses (2), let D be a discretisation in the sense of Definition 2.1. Let θ≥θ_D (defined by (16)). Let u be a solution of (11).

Then there exists C₃ >0, only depending on d, Ω, p, aand θ such that kuk^p_p,Π≤C₃kfk^p′

L^p′(Ω). (22)

Moreover, there exists one and only one solution u∈H_D to (11).

Proof. Letv=uin (11); applying (21) in Lemma 3.3, we get C₂kuk^p_p,Π≤

Z

f(x)u(x) dx.

We now apply H¨older’s inequality: we get Z

f(x)u(x) dx≤ kfk_Lp′

(Ω)kuk_L^p_(Ω).

We then use (3.1) in Lemma 3.1 and (19) to obtain that there existsC₄, only depending onθ,pand Ω, such that

kuk_L^p_(Ω)≤C₄kuk_p,Π. The last three inequalities yield (22).

The existence of the discrete solution follows, for instance by a topological degree argument.

Let us now assume thatuand ubare two solutions of (11). Then,hu, u−uib _D+hu,b bu−ui_D= 0, which yields

X

K∈M

|K|(a_K(∇_Ku)−a_K(∇_Ku))b · ∇_K(u−bu)

+ X

K∈M

X

σ∈EK

α_K,σ|σ|d_K,σ(sgn(R_K,σu)|R_K,σu|^p−1−sgn(R_K,σu)|Rb _K,σu|b^p−1) R_K,σ(u−bu) = 0.

Since

(sgn(RK,σu)|RK,σu|^p−1−sgn(RK,σbu)|RK,σu|b^p−1) (RK,σu−RK,σbu)≥0, and

(a_K(∇_Ku)−a_K(∇_Ku))b · ∇_K(u−u)b ≥0,

we get that both terms vanish. Thus, ∇_K(u−bu) = 0 and R_K,σ(u−u) = 0. We then get from theb inequality (21) that u−ub= 0.

4 Convergence of the scheme

The convergence of the scheme requires a compactness property which extends that of [19] proven in the case wherep= 2:

Lemma 4.1 (Weak discrete W^1,p compactness) Under hypotheses (2), let F be a family of dis- cretisations in the sense of Definition 2.1. For anyD ∈ F, let(Π_σ)_σ∈E be a family of linear mappings from H_D to R defined by (6). Assume that there exists θ > 0 such that for all D ∈ F, θ_D ≤θ, with θ_D defined by (16). Let(u_D)_D∈F be a family of piecewise constant functions such that:

• uD ∈HD for allD ∈ F,

• there existsC >0 with ku_Dk_p,Π≤C for all D ∈ F,

• there existsu∈L^p(Ω) with lim

hD→0kuD−uk_L^p_(Ω)= 0.

Then, the limit u belongs to W₀^1,p(Ω); moreover, defining ∇_Du_D by (8), the sequence (∇_Du_D)_D∈F weakly converges in L^p(Ω)^d to ∇u as hD → 0. Prolonging all functions by 0 outside of Ω, the convergence also holds in L^p(R^d)^d.

Proof. Thanks to (20), we get that, up to a sub-sequence, there exists some functionG∈L^p(R^d)^d such that ∇_Du_D weakly converges in L^p(R^d)^d to G as h_D → 0. Let us show that G = ∇u. Let ψ ∈C_c^∞(R^d)^d be given. Let us consider the term T₁^D defined by

T₁^D= Z

R^d

∇DuD(x)·ψ(x)dx;

this term may also be written:

T₁^D= X

K∈M

X

σ∈EK

|σ|(Πσu−uK)nK,σ·ψ_K, withψ_K = 1

|K|

Z

K

ψ(x)dx.

Let us then compareT₁^D with T₂^D= X

K∈M

X

σ∈EK

|σ|(Π_σu−u_K)n_K,σ·ψ_σ, whereψ_σ = 1

|σ|

Z

σ

ψ(x)dγ(x).

We get that

|T₁^D−T₄^D| ≤



 X

K∈M

X

σ∈EK

|σ|d_K,σ|Π_σu−u_K|^p d^p_K,σ





1/p

 X

K∈M

X

σ∈EK

|σ|d_K,σ|ψ_K−ψ_σ|^p′





1/p^′

, which leads to lim

hD→0(T₁^D−T₄^D) = 0.

Since

T₄^D =− X

K∈M

X

σ∈EK

|σ|uKn_K,σ·ψ_σ =− Z

R^d

ΠMuD(x)divψ(x)dx, we deduce that lim

hD→0T₄^D =−R

Rdu(x)divψ(x)dx. This proves that the function G∈L^p(R^d)^d is a.e.

equal to∇uinR^d. Sinceu= 0 outside of Ω, we get thatu∈W₀^1,p(Ω), and the uniqueness of the limit implies that the whole family ∇_Du_D weakly converges inL^p(R^d)^d to∇u ash_D →0.

For anyϕ∈C(Ω,R), we denote by P_Dϕ the element ofH_D defined by:

P_Dϕ= (ϕ(x_K))_K∈M, (23)

and we prove a consistency result of the discrete gradient, which was already used in [19] in the more general setting of the SUSHI scheme.

Lemma 4.2 (Discrete gradient consistency) Let D be a discretisation ofΩ in the sense of Defi- nition 2.1. and let (Π_σ)_σ∈E be a family of linear mappings from H_D toRdefined by (6). Assume that θ≥θ_D,Π (given by (17)). Then, for any function ϕ∈C²(Ω), there exists C₅ only depending ond, θ and ϕsuch that:

k∇_DP_Dϕ− ∇ϕk_(L^∞_(Ω))^d ≤C5h_D, (24) where ∇_D is defined by (8), and

kR_DP_Dϕk_L^∞_(Ω) ≤C₅h_D, (25) where R_D is defined by (9)-(10) and P_Dϕ is defined by (23).

Proof. From (8b), we have, for any K∈ M,

∇_KP_Dϕ= 1

|K|

X

σ∈EK

|σ|(Π_σP_Dϕ−ϕ(x_K))n_K,σ. Using definition (17), we can write, forσ ∈ E_K,

|ΠσPDϕ−ϕ(xσ)| ≤Cϕθh²_K, and on the other hand, we have

1

|K|

X

σ∈EK

|σ|(ϕ(x_σ)−ϕ(x_K))n_K,σ= 1

|K|

X

σ∈EK

|σ| ∇ϕ(x_K)·(x_σ−x_K) +h²_Kρ_K,σ n_K,σ,

where|ρ_K,σ| ≤C_ϕ withC_ϕ only depending onϕ. Thanks to (7) and to the regularity of the mesh, we get the existence of C₆, only depending on θ, such that

|∇_KP_Dϕ− ∇ϕ(x_K)| ≤h_KC₆C_ϕ.

From this last inequality, using Definition 9, we get the existence of C7, only depending on θ, such that

|R_K,σP_Dϕ| = 1

d_K,σ|Π_σP_Dϕ−ϕ(x_K)− ∇_KP_Dϕ·(x_σ−x_K)|

≤ h_KC_ϕC₇ which concludes the proof.

Theorem 4.1 (Convergence of the scheme) Under hypotheses (2), let F be a family of discreti- sations in the sense of Definition 2.1. For any D ∈ F, let (Π_σ)_σ∈E be a family defined by (6). We assume that,there existsθ >0 with, for all D ∈ F, θ≥θD,Π (see (17)).

For all D ∈ F, let u_D∈H_D be the unique solution of (11). Then, as h_D→0:

uD −→u∈W₀^1,p(Ω)for the strong topology of L^p(Ω), (26a) where u∈W₀^1,p(Ω) is the unique solution to (1); furthermore,

∇_Du_D−→ ∇u for the strong topology of L^p(Ω)^d and (26b) a(·,∇_Du_D)−→a(·,∇u) for the strong topology of(L^p′(Ω))^d. (26c)

Proof.

Step 1. Convergence of the approximate solution (proof of (26a)).

Thanks to (22) in Lemma 3.4, the family (ku_Dk_p,Π)_D∈F is bounded independently ofD. Therefore, thanks to inequality (19), we may apply Lemma 3.2. Thus, for any sequence (D_n)_n∈Nof discretisations in the family F such that h_Dn tends to 0 as n → ∞, there exist u ∈ L^p(Ω) such that, up to a sub- sequence,u_Dn →u inL^p(Ω) asn→ ∞. For short, we replace the indexD_n by nin the remainder of Step 1.

We then apply Lemma 4.1 and deduce that u ∈W₀^1,p(Ω) and that ∇nun → ∇u weakly in (L^p(Ω))^d as n→ ∞. Using Hypothesis (2e), we then get that the function x→a(x,∇_nu_n(x)) is bounded in (L^p′(Ω))^d; hence, there exists a sub-sequence of (D_n)_n∈N, again denoted (D_n)_n∈N, andA∈(L^p′(Ω))^d such thata(·,∇nu_n)→Aweakly in (L^p′(Ω))^d.

Let us now prove that prove that div(A−a(∇u)) = 0, and thatu is the weak solution of (1).

Let ψ∈C_c^∞(Ω). We introduceP_nψ as a test function in (11) (with P_n =P_Dn defined by (23)). We

get Z

Ω

a(x,∇_nu_n(x))· ∇_nP_nψ(x) dx+b(u_n, P_nψ) = Z

Ω

f(x)P_nψ(x) dx.

Thanks to the estimate (22) onu_n(in thek · k_1,p,Dnorm) and thanks to (20), the sequence (∇_nu_n)_n∈N

is bounded in L^p(Ω)^d. Hence, the sequence (Rnu)_n∈^N is bounded in L^p(Ω), which implies, using (25) in Lemma 4.2, that

n→∞lim b(u_n, P_nψ) = 0. (27)

Hence, using (24) in Lemma 4.2 and passing to the limit asn→+∞ in (11), we get that Z

Ω

A(x)· ∇ψ(x) dx= Z

Ω

f(x)ψ(x) dx. (28)

By density, this also holds for all ψ∈W₀^1,p(Ω). It remains to prove that Z

Ω

A(x)· ∇ψ(x) dx= Z

Ω

a(x,∇u(x))· ∇ψ(x) dx.

This is the object of the famous “Minty trick” [27]. Indeed, for anyψ∈C_c^∞(Ω), sincea satisfies (2d), the following inequality holds for anyn∈N:

Z

Ω

(a(x,∇nun(x))−a(x,∇nPnψ(x)))·(∇nun(x)− ∇nPnψ(x)) dx≥0, (29) On the other hand, thanks to the positivity of b(un, un) and becauseun verifies (11), we get

Z

Ω

a(x,∇_nu_n(x))· ∇_nu_n(x) dx≤ Z

Ω

a(x,∇_nu_n(x))· ∇_nu_n(x) dx+b(u_n, u_n) = Z

Ω

f(x)u_n(x) dx.

Passing to the limitn→+∞ (up to the considered sub-sequence), we thus get lim sup

n→∞

Z

Ω

a(x,∇_nu_n(x))· ∇_nu_n(x) dx≤ Z

Ω

f(x)u(x) dx= Z

Ω

A(x)· ∇u(x) dx,

thanks to (28) replacing ψ byu.

Now, thanks to the continuity on a with respect to its second argument (assumption (2b)) and to assumption (2e) , we obtain by the Lebesgue dominated convergence theorem that

a(·,∇_nP_nψ(x))→a(·,∇ψ) in L^p′(Ω) asn→+∞. (30)

Hence, passing to the limit asn→ ∞ in (29) leads to Z

Ω

(A(x)−a(x,∇ψ(x)))·(∇u(x)− ∇ψ(x)) dx≥0.

By density, this last inequality remains true for anyψ∈W₀^1,p(Ω). Takingψ=u+tϕwithϕ∈C_c^∞(Ω) and t >0, we get Z

Ω

(A(x)−a(x,∇u(x) +t∇ϕ(x)))· ∇ϕ(x) dx≥0, This gives, letting t→0, Z

Ω

(A(x)−a(x,∇u(x)))· ∇ϕ(x) dx≥0.

Changing ϕin−ϕ, we get

− Z

Ω

(A(x)−a(x,∇u(x)))· ∇ϕ(x) dx≥0.

This proves that R

Ω(A(x)−a(x,∇u(x)))· ∇ϕ(x) dx= 0 and therefore, with (28), thatuis the weak solution of (1). Then, using the uniqueness of the solution of (1), a classical argument gives that the whole sequence converges and therefore, this convergence holds for the whole family F as h_D → 0 which proves (26a).

Step 2. Strong convergence of the gradient (proof of (26b)).

We now prove the (strong) convergence of∇_Du_D to∇uin (L^p(Ω))^dash_D →0. We already remarked (thanks to the positivity ofb(u_D, u_D) and using thatu_D verifies (11)) that:

lim sup

hD→0

Z

Ω

a(x,∇_Du_D(x))· ∇_Du_D(x) dx≤ Z

Ω

f(x)u(x) dx= Z

Ω

a(x,∇u(x))· ∇u(x) dx.

Hence we get

lim sup

hD→0

Z

Ω

(a(x,∇_Du_D(x))−a(x,∇u(x)))·(∇_Du_D(x)− ∇u(x)) dx≤0.

Since (a(x,∇_Du_D)−a(x,∇u))·(∇_Du_D− ∇u)≥0 for a.e. x∈Ω, we then have

(a(·,∇_Du_D)−a(·,∇u))·(∇_Du_D− ∇u)→0 inL¹(Ω), (31) and therefore a.e. for a sub-sequence. Then, Lemma 4.3, (which states the “Leray-Lions trick” [27]

and which we give below for the sake of completeness) shows that ∇_Du_D → ∇u a.e. as h_D → 0, at least for the same sub-sequence. Since the family (∇_Du_D)_D∈F is bounded in L^p(Ω)^d, this a.e.

convergence implies the convergence inL^q(Ω)^dfor anyq∈[1, p). The convergence of the whole family (∇_Du_D)_D∈F to ∇u in L^q(Ω)^d classically follows, for any q ∈[1, p), as h_D → 0. The boundedness of the family (∇_Du_D)_D∈F L^p(Ω)^d) also entails that:

∇_Du_D converges to∇u weakly inL^p(Ω)^d ash_D →0.

In order to obtain the strong convergence of ∇_Du_D inL^p(Ω)^d (and not only inL^q(Ω)^d forq < p), we then remark that (31) gives:

hlimD→0

Z

Ω

a(x,∇_Du_D(x))· ∇_Du_D(x)dx= Z

Ω

a(x,∇u(x))· ∇u(x)dx. (32) Next, we notice that, for any sequence of discretisations, we can assume, up to a sub-sequence, the a.e.

convergence ofa(·,∇_Du_D)·∇_Du_Dtoa(·,∇u)·∇u; thus, sincea(·,∇_Du_D)·∇_Du_D ≥0 a.e., we also have

by Lemma 4.4 (which is again classical [27] and which we give below for the sake of completeness) a(·,∇_Du_D) · ∇_Du_D → a(·,∇u) · ∇u in L¹(Ω) as h_D → 0 (see Lemma 4.4). This L¹-convergence gives the equi-integrability of the family of functionsa(·,∇_Du_D)· ∇_Du_D, which, in turn, gives, thanks to (2c), that the family of functions |∇_Du_D|^p is equi-integrable. Finally, we obtain (using Vitali’s theorem) theL^p(Ω)^d convergence of∇_Du_D to∇u, ash_D→0.

Note that in the case a(x, ξ) =|ξ|^p−2ξ, a simple proof is possible since (32) gives the convergence of theL^p-norm of the approximate gradient to theL^p-norm of∇u (which is sufficient, thanks to the a.e.

convergence of “sub-sequences”, to obtain the L^p(Ω)^d convergence).

Step 3. Convergence of fluxes (proof of (26c)).

Since ∇_Du_D converges to ∇u in L^p(Ω)^d as h_D → 0, the convergence of a(·,∇_Du_D) to a(·,∇u) in L^p′(Ω)^d (namely assertion (26c)) follows classically from hypotheses (2b) and (2e) ona.

Lemma 4.3 (The ”Leray-Lions trick”) Let b be a continuous function from R^d to R^d such that (b(δ)−b(γ))·(δ−γ)>0 if δ, γ∈R^d, δ 6=γ. Let (β_n)_n∈N be a sequence in R^d and β ∈R^d such that (b(β_n)−b(β))·(β_n−β)→0 as n→ ∞. Then, β_n→β as n→ ∞.

Proof. We begin the proof with a preliminary remark. Let δ ∈R^d,δ 6= 0. We define the function h_δ from Rto Rby h_δ(s) = (b(β+sδ)−b(β))·δ. The hypothesis onbgives that h_δ is an increasing function since, for s > s^′, one has :

h_δ(s)−h_δ(s^′) = (b(β+sδ)−b(β+s^′δ))·δ >0.

We prove now, by contradiction, that lim_n→∞β_n =β. If the sequence (β_n)_n∈N does not converge to β, there exists ε > 0 and a subsequence, still denoted (βn)n∈N, such that|βn−β| ≥ε, for all n∈N. Then, we set δ_n= _|β^βn−β

n−β| and we can assume, up to a subsequence, that δ_n→δ asn→ ∞, for some δ∈R^dwith |δ|= 1. Taking s_n=|β_n−β|, we then have, since s_n≥ε:

(b(β_n)−b(β))·β_n−β sn

=h_δn(s_n)≥h_δn(ε) = (b(β+εδ_n)−b(β))·δ_n. Then, passing to the limit asn→ ∞,

0 = lim

n→∞

1

s_n(b(β_n)−b(β))·(β_n−β)≥(b(β+εδ)−b(β))·δ >0.

which is impossible.

Lemma 4.4 Let (Fn)_n∈^Nbe a sequence non-negative functions in L¹(Ω). LetF ∈L¹(Ω)be such that F_n→F a.e. in Ωand R

ΩF_n(x)dx→R

ΩF(x)dx, asn→ ∞. Then, F_n→F in L¹(Ω)as n→ ∞.

Proof. The proof of this lemma is very classical. Applying the Dominated Convergence Theorem to the sequence (F −F_n)⁺ leads to R

Ω(F(x)−F_n(x))⁺dx → 0 as n→ ∞. Then, since |F −F_n|= 2(F−F_n)⁺−(F−F_n), we conclude that F_n→F inL¹(Ω) asn→ ∞.

5 Numerical results

We consider the particular case Ω =]0,1[², and f given by the following: f(x) = 1 for a.e. x in a sub-domain of Ω and 0 elsewhere, as shown in the right part of Figure 5, and a is given by (3) for p∈]1,+∞[.

Figure 1: The mesh used (left), the functionf (f = 1 in the red part)

We then compute, for p = 1.3,1.6,2,3,6, an approximate solution of the solution of (11), using an under-relaxed fixed point method (consisting in computing|∇Du|^p−2 one iteration late). We then get the following results, on two meshes. On one hand, we use the mesh shown in the left part of Figure 5, on the other hand, we use a regular 80×80 square mesh. The results are provided in Figures 2, 3, 4, 5 and 6. We obtain the approximate solutions with a precision of 10⁻⁸ with a quite small number (less that 60) thanks to the under-relaxed fixed point method (we observed that without relaxation the method does not converge for p≥3).

Figure 2: Results for p= 1.3 on the irregular mesh (left) and on the regular mesh (right)

6 Conclusion and generalisations

We showed here the convergence of a cell centred scheme for the discretisation of nonlinear elliptic equations of the Leray-Lions type and showed its numerical efficiency on an example. The convergence analysis is performed by mimicking the functional analysis tools of [27] (such as the Minty-Browder trick and the Leray-Lions trick) in the discrete setting. It may be generalised to the case of pseudo- monotone operators, that isaof the form u7→a(·, u,∇u) with adequate assumptions ona(see [27]).

The convergence result can also be extended, using the discrete functional tools which are introduced here and the results of [15], to handle right hand sides f ∈ W^−1,p′. Both of these extensions are presented in [13] in the framework of the mixed finite volume scheme.

Let us also remark that the SUCCES scheme which we applied here to the discretisation of the Leray- Lions operator is closer, in its formulation, to a low order non conforming finite element scheme than

Figure 3: Results for p= 1.6 on the irregular mesh (left) and on the regular mesh (right)

Figure 4: Results forp= 2 on the irregular mesh (left) and on the regular mesh (right)

Figure 5: Results forp= 3 on the irregular mesh (left) and on the regular mesh (right)

Figure 6: Results forp= 6 on the irregular mesh (left) and on the regular mesh (right) to finite volume scheme: indeed, even tough global conservativity of the fluxes is ensured, the scheme may not in general be written as a system of discrete balance laws over the discretisation cells, such as for the “classical” finite volume schemes, such as the two point flux scheme on admissible meshes, [17, chapter 3], the previously mentioned DDFV scheme [12] and mixed finite volume scheme [14], and the SUSHI scheme in its hybrid form [18, 19]. A scheme adapted from SUCCES, localising the fluxes on half edges as in the O scheme [1, 2] was recently introduced [4]: this scheme is a classical finite volume scheme in the sense that it may be written as a system of discrete mass balances and that local numerical fluxes are conservative. Nevertheless, it preserves the main properties of the SUCCES scheme (cell centred unknowns, consistent gradient, compactness properties) and therefore, the convergence theory which was presented here should also extend to this latter scheme (which, at the present time, is only written in the two dimensional setting).

Finally, we again stress that the tools used here essentially mimic the functional analysis tools of [27].

Since the obtained results include the strong convergence of the approximate solution, gradient, and fluxes, they may also be used for nonlinear evolution problems such as the level set equation [20].

References

[1] I. Aavatsmark, T. Barkve, O. Boe, and T. Mannseth. Discretization on unstructured grids for inhomogeneous, anisotropic media. part i: Derivation of the methods. SIAM Journal on Sc.

Comp., 19:1700–1716, 1998.

[2] I. Aavatsmark, T. Barkve, O. Boe, and T. Mannseth. Discretization on unstructured grids for inhomogeneous, anisotropic media. part ii: Discussion and numerical results. SIAM Journal on Sc. Comp., 19:1717–1736, 1998.

[3] L. Agelas, D.A. Di Pietro, and R. Masson. A symmetric and coercive finite volume scheme for multiphase porous media flow problems with applications in the oil industry. In R. Eymard and J.-M. H´erard, editors,Finite Volumes for Complex Applications V, pages 35–51. Wiley, 2008.

[4] L. Agelas, R. Eymard, and R. Herbin. A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. submitted, see also http://hal.archives-ouvertes.fr/hal- 00350139/fr/.

[5] B. Andreianov, F. Boyer, and F. Hubert. Finite volume schemes for thep-Laplacian on Cartesian meshes. M2AN Math. Model. Numer. Anal., 38(6):931–959, 2004.

[6] B. Andreianov, F. Boyer, and F. Hubert. Besov regularity and new error estimates for finite volume approximations of thep-Laplacian. Numer. Math., 100(4):565–592, 2005.

[7] B. Andreianov, F. Boyer, and F. Hubert. On the finite-volume approximation of regular solutions of thep-Laplacian. IMA J. Numer. Anal., 26(3):472–502, 2006.

[8] B. Andreianov, F. Boyer, and F. Hubert. Discrete duality finite volume schemes for Leray-Lions- type elliptic problems on general 2D meshes. Numer. Methods Partial Differential Equations, 23(1):145–195, 2007.

[9] J. W. Barrett and W. B. Liu. Finite element approximation of the p-Laplacian. Math. Comp., 61(204):523–537, 1993.

[10] F. Boyer and F. Hubert. Finite volume method for 2d linear and nonlinear elliptic problems with discontinuities. SIAM J. Numer. Anal., 46(6):3032–3070, 2008.

[11] S.-S. Chow. Finite element error estimates for nonlinear elliptic equations of monotone type.

Numer. Math., 54(4):373–393, 1989.

[12] K. Domelevo and P. Omnes. A finite volume method for the laplace equation on almost arbitrary two-dimensional grids. M2AN Math. Model. Numer. Anal., 39(6):1203–1249, 2005.

[13] J. Droniou. Finite volume schemes for fully non-linear elliptic equations in divergence form.

M2AN Math. Model. Numer. Anal., 40(6):1069–1100, 2006.

[14] J. Droniou and R. Eymard. A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math., 105(1):35–71, 2006.

[15] J. Droniou and T. Gallou¨et. Finite volume methods for convection-diffusion equations with right- hand side inH⁻¹. M2AN Math. Model. Numer. Anal., 36(4):705–724, 2002.

[16] C. Ebmeyer and W. B. Liu. Finite element approximation of the fast diffusion and the porous medium equations. SIAM J. Numer. Anal., 46(5):2393–2410, 2008.

[17] R. Eymard, T. Gallou¨et, and R. Herbin. Finite volume methods. In P. G. Ciarlet and J.-L.

Lions, editors, Techniques of Scientific Computing, Part III, Handbook of Numerical Analysis, VII, pages 713–1020. North-Holland, Amsterdam, 2000.

[18] R. Eymard, T. Gallou¨et, and R. Herbin. A new finite volume scheme for anisotropic diffusion problems on general grids: convergence analysis. C. R., Math., Acad. Sci. Paris, 344(6):403–406, 2007.

[19] R. Eymard, T. Gallou¨et, and R. Herbin. Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes, sushi: a scheme using stabilisation and hybrid interfaces. to appear in IMAJNA, see also http://hal.archives-ouvertes.fr/docs/00/21/18/28/PDF/suchi.pdf , 2008.

[20] R. Eymard, A. Handloviˇcov´a, and K. Mikula. Convergence of a finite volume method to a solution of the level set equation. work in progress, 2009.

[21] R. Eymard and R. Herbin. A new colocated finite volume scheme for the incompressible Navier- Stokes equations on general non matching grids. C. R. Math. Acad. Sci. Paris, 344(10):659–662, 2007.

[22] M. Feistauer and V. Sobot´ıkov´a. Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. RAIRO Mod´el. Math. Anal. Num´er., 24(4):457–500, 1990.

[23] M. Feistauer and A. ˇZen´ıˇsek. Finite element solution of nonlinear elliptic problems. Numer.

Math., 50(4):451–475, 1987.

[24] M. Feistauer and A. ˇZen´ıˇsek. Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math., 52(2):147–163, 1988.

[25] R. Glowinski and J. Rappaz. Approximation of a nonlinear elliptic problem arising in a non- Newtonian fluid flow model in glaciology. M2AN Math. Model. Numer. Anal., 37(1):175–186, 2003.

[26] F. Hermeline. Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Engrg., 192(16-18):1939–1959, 2003.

[27] J. Leray and J.-L. Lions. Quelques r´esultats de Viˇsik sur les probl`emes elliptiques nonlin´eaires par les m´ethodes de Minty-Browder. Bull. Soc. Math. France, 93:97–107, 1965.

[28] A. ˇZen´ıˇsek. The finite element method for nonlinear elliptic equations with discontinuous coeffi- cients. Numer. Math., 58(1):51–77, 1990.