Caterina Calgaro
email: caterina.calgaro@univ-lille.fr

Clément Cancès

Emmanuel Creusé

Discrete Gagliardo-Nirenberg inequality and application to the finite volume approximation of a convection-diffusion equation with a Joule effect term

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

Discrete Gagliardo-Nirenberg inequality and application to the finite volume approximation of a convection-diffusion equation with a Joule effect term

Introduction

Variable density low-Mach models arise in a wide range of physical phenomena, in which the sound wave speed is much faster than the convective characteristic of the fluids. In the case of a calorically perfect gas, an asymptotic expansion of the variables with respect to the Mach number in the compressible Navier-Stokes equation leads to a low-Mach system [START_REF] Majda | The derivation and numerical solution of the equations for zero Mach number combustion[END_REF]. In the particular configuration where the dynamic viscosity of the fluid can be explicitly given as a specific function of the temperature (see [START_REF] Bresch | Effect of density dependent viscosities on multiphasic incompressible fluid models[END_REF][START_REF] Bresch | Two-velocity hydrodynamics in fluid mechanics: Part I. Well posedness for zero Mach number systems[END_REF]), a change of variables can be used in order to obtain a divergence-free velocity system. In that case, the mass conservation equation is reformulated in term of the temperature u instead of the density 1{u, and contains a nonlinear term called the "Joule effect term" [8, Section 1] in analogy with systems accounting for energy dissipation stemming from electric currents, see for instance [START_REF] Jüngel | Quasi-hydrodynamic semiconductor equations[END_REF]. Results on local and global well-posedness of this system have been recently obtained under some smallness assumptions on the initial data, see e.g. [START_REF] Huang | On the strong solution of the ghost effect system[END_REF][START_REF] Calgaro | Approximation by an iterative method of a low-Mach model with temperature dependent viscosity[END_REF], still based on a formulation using the temperature as a primary variable.

From the numerical point of view, a combined finite volume -finite element scheme was proposed in [START_REF] Calgaro | A combined finite volume-finite element scheme for a low-Mach system involving a Joule term[END_REF] to simulate such a model in terms of temperature, velocity and pressure. The method is based on a time splitting, in which the first step consists in solving the mass conservation by an ad-hoc finite volume scheme. In dimensionless form, the mass conservation equation, set in a subdomain Ω of R 2 and for positive times, writes B t u `∇ ¨pu vq `λ |∇u| 2 ´λu ∆u " 0,

where v is a given velocity field computed in the other step of the splitting algorithm.

It is complemented by homogeneous Neumann boundary conditions and an initial datum u 0 . It has been proved in [START_REF] Calgaro | A combined finite volume-finite element scheme for a low-Mach system involving a Joule term[END_REF] that the scheme referred to as SD moy J up therein preserves a discrete maximum principle property, as imposed by the physics of the problem. This paper is devoted to the numerical analysis of this finite volume numerical scheme for the approximation of the temperature, solution of [START_REF] Andreianov | A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs[END_REF].

Here, we rigorously establish the convergence of the finite volume solution towards the exact continuous one in the particular case of successively refined two-dimensional cartesian grids. As often, the method consists in deriving some a priori estimates on the numerical solution in order to establish the existence of discrete solutions to the scheme. Then, with the help of further estimates and of some compactness properties, the limits are proven to be the weak solutions of the equation. The originality of this contribution comes from the presence of the Joule effect term in the equation, leading to some specific difficulties.

Several times in our proof, we will make use of a discrete version of the inequality

}∇u} 2 L 4 pΩq 2 ď C GN }∇ 2 u} L 2 pΩq 2ˆ2 }u} L 8 pΩq (2)
which has not been established before up to our knowledge. At the continuous level, some classical Gagliardo-Nirenberg interpolation inequalities for intermediate derivatives in R n have been established in the seminal papers of E. Gagliardo [START_REF] Gagliardo | Ulteriori proprietà di alcune classi di funzioni in più variabili[END_REF] and L.

Nirenberg [START_REF] Nirenberg | On elliptic partial differential equations[END_REF]. The following particular case of these results is stated in [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF]Theorem 1.2] in the following form.

Theorem 1.1. If 1 ď q ď 8, 1 ď r ă 8, j, k P N, j ă k and

1 p " j kr `k ´j kq ,
then there exists a constant C GN independent of u such that }∇ j u} k L p pR n q ď C GN }∇ k u} j L r pR n q }u} k´j L q pR n q @u P L q pR n q X W k,r pR n q.

This result holds in the particular case j " 1 and k " 2, so that if 2 p " 1 r `1 q , we have: }∇u} L p pR n q ď C GN }∇ 2 u} 1{2 L r pR n q }u} 1{2 L q pR n q @u P L q pR n q X W 2,r pR n q.

(3)

Up to the fact that (2) holds on a bounded domain Ω (which does not yield particular difficulties), ( 2) is a particular case of [START_REF] Bouchut | Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data[END_REF]. In what follows, we refer to (3) as a second order Gagliardo-Nirenberg inequality since the highest order of differentiation is k " 2.

The first goal of this paper is to establish a discrete version of (3) in the particular case of some piecewise constant discrete functions defined on a cartesian grid. As far as we know, only first order discrete (Sobolev-)Gagliardo-Nirenberg inequalities are available in the literature so far, see [START_REF] Bouchut | Finite volume schemes for the approximation via characteristics of linear convection equations with irregular data[END_REF][START_REF] Bessemoulin-Chatard | On discrete functional inequalities for some finite volume schemes[END_REF]. This new discrete estimate looks to us as a key element for new contributions in the field of the numerical analysis of partial differential equations. The outline of the paper is the following. Section 2 introduces the discrete setting of the problem: the meshes, the associated discrete functional spaces and the discrete difference operators on these spaces. Section 3 is devoted to the second order discrete Gagliardo-Nirenberg inequality which is obtained following the lines of the continuous case detailed in [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF], leading to Theorem 3.7. Then, Section 4 presents the finite volume scheme SD moy J up previously introduced in [START_REF] Calgaro | A combined finite volume-finite element scheme for a low-Mach system involving a Joule term[END_REF] for the approximation of the convection-diffusion equation involving a Joule effect term, which is here formulated in the case of a two-dimensional Cartesian grid. We infer from the discrete maximum principle established in Section 5 that the scheme admits a unique solution, and further estimates of energy type are derived under some smallness assumption on the data. Section 6 then addresses the convergence of the finite volume solution towards the continuous one as the discretization parameters tend to 0, cf. Theorem 4.4. Some mostly elementary properties of the discrete operators are finally collected in Appendix for the ease of reading.

Discrete setting

In this section, we introduce the discrete framework such as the cartesian meshes, some discrete functional spaces as well as some differential operators needed for the following of the paper. We consider successively the 1D case and then the 2D one.

The case d " 1

Let I "sx, xr be an open set of R, consisting in a union of cells M defined by (see Figure 1):

M " tC i "sx i´1 2 , x i`1 2 r, i P v1, N wu, with N P N ˚. We also define the set of shifted cells:

x M " tC i`1 2 "sx i , x i`1 r, i P v0, N wu.

We denote |I| " x ´x the length of I and we define x 0 " x 1 2 " x and x N `1 2 " x N `1 " x. Let x i be the center and h i " x i`1 2 ´xi´1 2 the length of C i for i P v1, N w. Let x i`1 2 be the center and h i`1 2 " x i`1 ´xi the length of C i`1 2 for i P v0, N w. The spaces H M pIq, p H M pIq and p H 0 M pIq are respectively defined by:

x0 " x 1 2 " x x i´1 2 x i`1 2 x " x N `1 2 " x N `1 Ci xi ' xi`1 C i`1 2 ' x1 ' x N '
H M pIq " tv P L 8 pIq | v |Ci " v i P R, i P v1, N wu, p H M pIq " tv P L 8 pIq | v |C i`1 2 " v i`1 2 P R, i P v0, N wu, p H 0 M pIq " tv P p H M pIq | v 1 2 " v N `1 2 " 0u.
The discrete operator δ x is defined from H M pIq in p H 0 M pIq by:

δ x vpxq |C i`1 2 " $ & % v i`1 ´vi h i`1 2 for i P v1, N ´1w, 0 for i " 0 and i " N.
Similarly, the discrete gradient operator δ x is defined from p H M pIq in H M pIq by:

δ x vpxq |Ci " v i`1 2 ´vi´1 2 h i for i P v1, N w.
The discrete second-order derivative operator δ xx is defined from H M pIq in H M pIq by: δ xx v " pδ x ˝δx qv.

Now the interpolation operator π

x is defined from H M pIq in p H M pIq by:

pπ x vq C i`1 2 " $ ' & ' % v i `vi`1 2 for i P v1, N ´1w, v 1 for i " 0, v N for i " N,
and the interpolation operator π x is defined from p H M pIq in H M pIq by:

pπ x vq Ci " v i´1 2 `vi`1 2 2 for i P v1, N w.

2.2

The case d ą 1

Meshes and discrete functional spaces

We consider Ω a connected subset of R d consisting in a union of rectangles (d " 2) or parallelepipeds (d " 3q, possibly non-uniform. The edges (or faces) of these rectangles (or parallelepipeds) are assumed to be orthogonal to the canonical basis vectors. All the notations are given in the case d " 2, but they can be generalized to the case d " 3.

xi yj x " x0 " x 1 2 x i´1 2 x i`1 2 hi x " x N `1 2 " x N `1 y0 " y 1 2 " y y j´1 2 y j`1 2 kj y M `1 " y M `1 2 " y Ci,j σ i`1 2 ,j P E V σ i,j`1 2 P E H C i`1 2 ,j C i,j`1 2 C i`1 2 ,j`1 2
Let Ω "sx, xrˆsy, yrĂ R 2 be the set of the grid cells M defined by (see Figure 2):

M " ! C i,j " sx i´1 2 , x i`1 2 rˆsy j´1 2 , y j`1 2 r, i P v1, N w, j P v1, M w ) ,
with N, M P N ˚. We also define the set of shifted cells in the x-direction:

x M " ! C i`1 2 ,j " sx i , x i`1 rˆsy j´1 2 , y j`1 2 r, i P v0, N w, j P v1, M w ) ,
the set of shifted cells in the y-direction:

Ă M " ! C i,j`1 2 " sx i´1 2 , x i`1 2 rˆsy j , y j`1 r, i P v1, N w, j P v0, M w ) ,
and the set of shifted cells in the x-y-directions:

Ď M " ! C i`1 2 ,j`1 2 " sx i , x i`1 rˆsy j , y j`1 r, i P v0, N w, j P v0, M w ) .
Similarly to the 1D case, we define:

h i`1 2 " x i`1 ´xi for i P v0, N w, h i " x i`1 2 ´xi´1 2 for i P v1, N w,
k j`1 2 " y j`1 ´yj for j P v0, M w, k j " y j`1 2 ´yj´1 2 for j P v1, M w.

We introduce the following functional spaces:

H M pΩq " tv P L 8 pΩq | v |Ci,j " v i,j P R, i P v1, N w, j P v1, M wu, p H M pΩq " tv P L 8 pΩq | v |C i`1 2 ,j " v i`1 2 ,j P R, i P v0, N w, j P v1, M wu, p H 0 M pΩq " tv P p H M pΩq | v 1 2 ,j " v N `1 2 ,j " 0, j P v1, M wu, r H M pΩq " tv P L 8 pΩq | v |C i,j`1 2 " v i,j`1 2 P R, i P v1, N w, j P v0, M wu, r H 0 M pΩq " tv P r H M pΩq | v i, 1 2 " v i,M `1 2 " 0, i P v1, N wu, H M pΩq " p H M pΩq ˆr H M pΩq, H 0 M pΩq " p H 0 M pΩq ˆr H 0 M pΩq, s H M pΩq " tv P L 8 pΩq | v |C i`1 2 ,j`1 2 " v i`1 2 ,j`1 2 P R, i P v0, N w, j P v0, M wu, s H 0,x M pΩq " tv P s H M pΩq, | v |C 1 2 ,j`1 2 " v |C N `1 2 ,j`1 2 " 0, j P v0, M wu, s H 0,y M pΩq " tv P s H M pΩq, | v |C i`1 2 , 1 2 " v |C i`1 2 ,M `1 2 " 0, i P v0, N wu, s H 0,0 M pΩq " s H 0,x M pΩq X s H 0,y M pΩq.

Discrete differential operators

The discrete operator δ x is defined from H M pΩq in p H 0 M pΩq for j P v1, M w (respectively from r H M pΩq in s H M pΩq for j P v0, M w `1 2 ) by:

δ x vpx, yq |C i`1 2 ,j " $ & % v i`1,j ´vi,j h i`1 2
for i P v1, N ´1w, 0 for i " 0 and i " N.

Similarly, the discrete operator δ y is defined from H M pΩq in r H 0 M pΩq for i P v1, N w (respectively from p H M pΩq in s H M pΩq for i P v0, N w `1 2 ) by :

δ y vpx, yq |C i,j`1 2 " $ & % v i,j`1 ´vi,j k j`1 2
for j P v1, M ´1w, 0 for j " 0 and j " M.

The discrete operator δ x is defined from p H M pΩq in H M pΩq for j P v1, M w (respectively from s H M pΩq in r H M pΩq for j P v0, M w `1 2 ) by:

δ x vpx, yq |Ci,j " v i`1 2 ,j ´vi´1 2 ,j h i for i P v1, N w.
Similarly, the discrete operator δ ẙ is defined from r H M pΩq in H M pΩq for i P v1, N w (respectively from s H M pΩq in p H M pΩq for i P v0, N w `1 2 ) by : δ ẙ vpx, yq |Ci,j " v i,j`1 2 ´vi,j´1 2 k j for j P v1, M w.

Then, the discrete gradient operator ∇ h is defined by:

∇ h : H M pΩq Ñ H 0 M pΩq v Þ Ñ ∇ h v " ˆδx v δ y v ˙.
With a slight abuse of notation, we also denote in what follows by

∇ h : H 0 M pΩq Ñ H M pΩq ˆs H M pΩq ˆs H M pΩq ˆHM pΩq v " ˆv1 v 2 ˙Þ Ñ ∇ h v " ˆδx v 1 δ y v 1 δ x v 2 δ ẙ v 2 ˙. (4)
The operators δ xx " δ x ˝δx and δ yy " δ ẙ ˝δy are defined on H M pΩq, so that the discrete Laplace operator ∆ h is defined by:

∆ h : H M pΩq ÝÑ H M pΩq v ÝÑ ∆ h v " pδ xx `δyy qv.
Finally, it remains to define the cross-derivatives by the operators δ yx and δ xy respectively defined from H M pΩq in s H 0,0 M pΩq by:

δ yx v " δ xy v " pδ y ˝δx qv " pδ x ˝δy qv, (5)
and the discrete Hessian matrix ∇ 2 h is defined for any v P H M pΩq by:

∇ 2 h v " ˆδxx v δ yx v δ xy v δ yy v ˙.

Discrete interpolation operators

Some discrete interpolation operators are needed in order to pass from a given grid to another one, similarly to the ones given in subsection 2.1 for d " 1. The interpolation operator π x is defined from H M pΩq in p H M pΩq for j P v1, M w (respectively from r H M pΩq in s H M pΩq for j P v0, M w `1 2 ) by:

pπ x vq C i`1 2 ,j " $ ' & ' % v i,j `vi`1,j 2 for i P v1, N ´1w, v 1,j for i " 0, v N,j for i " N.
Similarly, the interpolation operator π y is defined from H M pΩq in r H M pΩq for i P v1, N w (respectively from p H M pΩq in s H M pΩq for i P v0, N w `1 2 ) by:

pπ y vq C i,j`1 2 " $ ' & ' % v i,j `vi,j`1 2 for j P v1, M ´1w, v i,1 for j " 0, v i,M
for j " M.

The interpolation operator π x is defined from p H M pΩq in H M pΩq for j P v1, M w (respectively from s H M pΩq in r H M pΩq for j P v0, M w `1 2 ) by:

pπ x vq Ci,j " v i´1 2 ,j `vi`1 2 ,j 2
for i P v1, N w.

Similarly, the interpolation operator π ẙ is defined from r H M pΩq in H M pΩq for i P v1, N w (respectively from s H M pΩq in p H M pΩq for i P v0, N w `1 2 ) by: pπ ẙ vq Ci,j " v i,j´1 2 `vi,j`1 2 2 for j P v1, M w.

In the following of the paper, the discrete differential and interpolation operators fulfill some discrete properties, which are collected in Appendix A.

Norm definitions

Let p P R, p ě 1. For any v P L p pΩq, the L p norm is denoted:

}v} L p pΩq " ˆżΩ |v| p dx ˙1{p . (6)
The norm in the case p " 8 means the essential supremum over Ω. For any v " pv i q 1ďiďd P pL p pΩqq d , we define:

}v} L p pΩq " ˜d ÿ i"1 }v i } p L p pΩq ¸1{p ,
and for any v " pv i,j q 1ďi,jďd P pL p pΩqq dˆd , the L p norm of v is defined by:

}v} p L p pΩq " d ÿ i,j"1 }v i,j } p L p pΩq . ( 7
)
3 The discrete Gagliardo-Nirenberg inequality

The 1D case

The goal of this subsection is to establish the discrete Gagliardo-Nirenberg inequality corresponding to the discrete 1D counterpart of (3):

Theorem 3.1. Let v P H M pIq, 1 ď p, r ă 8 and 1 ď q ď 8 such that

2 p " 1 r `1 q .
Then there exists C GN independent of v such that:

}δ x v} L p pIq ď C GN }δ xx v} 1{2 L r pIq }v} 1{2 L q pIq . (8)
We start to prove Theorem 3.1, following from the discrete point of view the work of [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF] corresponding to the continuous case. We first establish some Lemma.

Lemma 3.2. For any v P H M pIq, any J Ă I and for any 1 ď p ď 8 we have:

ˇˇˇˇv ´1 |J| ż J vpxq dx ˇˇˇˇL p pJq ď 2 inf cPR }v ´c} L p pJq .
Proof. The proof is exactly the same as the one of Lemma 3.1 of [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF] since H M pIq Ă L p pJq, 1 ď p ď 8.

Lemma 3.3. Let r P R, r ě 1, v P H M pIq and J Ă I such that:

ż J δ x vpxq dx " 0. ( 9
)
Then for any x P J we have:

|δ x vpxq| ď 2 }δ xx v} L r pJq |J| r´1 r . ( 10
)
Proof. We note J "sα, βr, where α P C iα`1 2 and β P C i β `1 2 with pi α , i β q P v0, N w 2 , i α ď i β (see Figure 3). Without loss of generality, we suppose i α ă i β . We define hiα`1 2 " x iα`1 ´α, hi β `1 2 " β ´xi β and hk`1 2 " h k`1 2 , k P vi α `1, i β ´1w.

' x1 ' x2 xi α ' xi α`1 x iα`1 2 ' α β xi β ' xi β `1 ' x i β `1 2 ' x N
The property [START_REF] Cancès | A variational finite volume scheme for Wasserstein gradient flows[END_REF] can consequently be written as:

i β ÿ k"iα hk`1 2 pδ x vq k`1 2 " 0. (11)
Let x P J and j P vi α , i β w such that |pδ x vq j`1 2 | " }δ x v} L 8 pJq . We have:

|δ x vpxq| ď |pδ x vq j`1 2 |. (12)
Considering now i P vi α , i β w such that:

|pδ x vq j`1 2 ´pδ x vq i`1 2 | " max kPviα,i β w |pδ x vq j`1 2 ´pδ x vq k`1 2 |,
we have from [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF]:

|pδ x vq j`1 2 | ď |pδ x vq j`1 2 ´pδ x vq i`1 2 | ď j ÿ k"i`1 |pδ x vq k`1 2 ´pδ x vq k´1 2 | ď i β ÿ k"iα`1 |pδ x vq k`1 2 ´pδ x vq k´1 2 |. (13)
We suppose first that α ą x iα`1 2 and β ă x i β `1 2 , like in Figure 3. We introduce:

hiα`1 " hiα`1 2 `hiα`1 2 , hi β " hi β `1 2 `hi β 2 , hk " h k for k P vi α `2, i β ´1w.
We know that h iα`1 ď 2 hiα`1 and h i β ď 2 hi β , and consequently:

i β ÿ k"iα`1 ˇˇpδxvq k`1 2 ´pδ x vq k´1 2 ˇˇď 2 i β ÿ k"iα`1 |ppδ x ˝δx qvq k | hk .
By the Hölder inequality we get:

i β ÿ k"iα`1 ˇˇpδxvq k`1 2 ´pδ x vq k´1 2 ˇď 2 ˜iβ ÿ k"iα`1 |ppδ x ˝δx qvq k | r hk ¸1{r ˜iβ ÿ k"iα`1 hk ¸r´1 r " 2 }δ xx v} L r pJq |J| r´1 r . (14)
From ( 12), ( 13) and ( 14) we get [START_REF] Deimling | Nonlinear functional analysis[END_REF] in the case α ą x iα`1 2 and β ă x i β `1 2 . Then, in the case α ă x iα`1 2 and β ą x i β `1 2 , we introduce:

hiα " x iα`1 2 ´α, hi β `1 " β ´xi β `1 2 , hk " h k for k P vi α `1, i β w.
This time, introducing J " rx iα`1 2 , x i β `1 2 s, we obtain by the Hölder inequality:

i β ÿ k"iα`1 |pδ x vq k`1 2 ´pδ x vq k´1 2 | ď ˜iβ ÿ k"iα`1 |ppδ x ˝δx qvq k | r h k ¸1{r ˜iβ ÿ k"iα`1 h k ¸r´1 r " }δ xx v} L r pJq |J| r´1 r ď }δ xx v} L r pJq |J| r´1 r . (15)
From ( 12), ( 13) and ( 15) we get [START_REF] Deimling | Nonlinear functional analysis[END_REF] in the case α ă x iα`1 2 and β ą x i β `1 2 . Finally, in the two last cases (respectively

α ă x iα`1 2 , β ă x i β `1 2 and α ą x iα`1 2 , β ą x i β `1 2 ),
we proceed similarly and we obtain [START_REF] Deimling | Nonlinear functional analysis[END_REF]. The proof is complete.

Lemma 3.4. Let r P R, r ě 1, v P H M pIq and J Ă I such that ż J vpxq dx " 0. ( 16
)
Then for any x P J we have:

|vpxq| ď 2 }δ x v} L r pJq |J| r´1 r . ( 17
)
Proof. The proof is very similar to the one of Lemma 3.3 and based on the same arguments. We note J "sα, βr, where α P C iα and β P C i β with pi α , i β q P v1, N w 2 , i α ď i β . Without loss of generality, we suppose i α ă i β . We define hiα " x iα`1 2 ´α, hi β " β ´xi β ´1 2 and hk " h k , k P vi α `1, i β ´1w. The property ( 16) can be written as:

i β ÿ k"iα hk v k " 0. ( 18
)
Now, let x P J and j P vi α , i β w such that |v j | " }v} L 8 pJq . We have:

|vpxq| ď |v j |. (19)
Considering now i P vi α , i β w such that |v j ´vi | " max kPviα,i β w |v j ´vk |, we have from ( 18):

|v j | ď |v j ´vi | ď j ÿ k"i`1 |v k ´vk´1 | ď i β ÿ k"iα`1 |v k ´vk´1 |. (20)
We suppose first that α ą x iα and β ă x i β . We introduce:

hiα`1 2 " hiα `hiα`1 2 2 , hi β ´1 2 " hi β `hi β ´1 2 2 , hk`1 2 " h k`1 2 for k P vi α `1, i β ´2w.
We know that h iα`1 2 ď 2 hiα`1 2 and h i β ´1 2 ď 2 hi β ´1 2 . Consequently:

i β ÿ k"iα`1 |v k ´vk´1 | ď 2 i β ÿ k"iα`1 ˇˇpδxvq k´1 2 ˇˇh k´1 2 .
By the Hölder inequality we get:

i β ÿ k"iα`1 |v k ´vk´1 | ď 2 ˜iβ ÿ k"iα`1 ˇˇpδxvq k´1 2 ˇˇr hk´1 2 ¸1{r ˜iβ ÿ k"iα`1 hk´1 2 ¸r´1 r " 2 }δ x v} L r pJq |J| r´1 r . (21)
From ( 19), ( 20) and ( 21) we get [START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic PDEs-application to a turbulence model[END_REF] in the case α ą x iα and β ă x i β .

For the three other cases (α ă x iα and β ă x i β ; α ą x iα and β ą x i β ; α ă x iα and β ą x i β ), we proceed in the same way, similarly to Lemma 3.3.

Lemma 3.5. Let p, q, r P R, p ě 1, q ě 1, r ě 1, J Ă I. For any v P H M pIq, there exists C independent of v such that

}δ x v} L p pJq ď C ´|J| 1`1 p ´1 r }δ xx v} L r pJq `|J| ´1`1 p ´1 q }v} L q pJq ¯.
Proof. The proof is similar to the one of Lemma 3.2 in [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF], and we give it here for completeness. First we introduce

v " 1 |J| ż J vpxq dx.
From Lemma 3.2, we have:

}v ´v} L q pJq « inf cPR }v ´c} L q pJq ,
so that we may assume v " 0. We denote d " 1 |J| ż J δ x vpxq dx and X 0 the center of J, and we define

r vpxq " vpxq ´dpx ´X0 q, ( 22
) so that ż J r vpxq dx " 0, (23) and ż
J δ x r vpxq dx " 0. (24)
From ( 24) and Lemma 3.3, we get:

}δ x r v} L p pJq ď 2 |J| r´1 r `1 p }δ xx v} L r pJq . (25)
From ( 23), Lemma 3.4 and (25), we get:

}r v} L q pJq ď 4 |J| 1`1 q `r´1 r }δ xx v} L r pJq . (26)
Finally we have:

}δ x v} L p pJq (22) ď }δ x r v} L p pJq `}d} L p pJq " }δ x r v} L p pJq `}1} L p pJq ¨}dpx ´X0 q} L q pJq }x ´X0 } L q pJq (25) À |J| 1`1 p ´1 r }δ xx v} L r pJq `|J| ´1`1 p ´1 q }dpx ´X0 q} L q pJq (22) ď |J| 1`1 p ´1 r }δ xx v} L r pJq `|J| ´1`1 p ´1 q p}v} L q pJq `}r v} L q pJq q (26) À |J| 1`1 p ´1 r }δ xx v} L r pJq `|J| ´1`1 p ´1 q }v} L q pJq .
The proof is complete.

Lemma 3.6. Let v P H M pIq and 1 ď p, r ă 8, 1 ď q ď 8 such that

2 p " 1 r `1 q .
Then, there exists a sequence of open intervals pI k q, which covers I, such that:

|I k | 1`1 p ´1 r }δ xx v} L r pI k q " |I k | ´1`1 p ´1 q }v} L q pI k q , ÿ k χ I k ď 4.
Proof. The proof is exactly the same as the one of Lemma 3.3 in [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF]. We only mention that we need the values of r and q not to be equal to `8, so that the functions ω x and α x remain continuous, since in the proof we have to replace C 8 c pRq by H M pIq.

Proof. (of Theorem 3.1)

First, we consider 1 ď p n , q n , r n ă 8 such that 2 p n " 1 r n `1 q n . Following the proof of Lemma 3.4 in [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF] and using previous Lemma 3.5 and 3.6 (which respectively correspond to the discrete versions of Lemma 3.2 and 3.3 in [START_REF] Fiorenza | Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks[END_REF]), we obtain: Now it remains to make q n tend towards `8 to obtain (8) in the case q " 8.

}δ x v} pn L pn pIq À }δ xx v} pn 2 L rn pIq }v} pn 2 L qn pIq , (27)

The 2D case

The goal of this subsection is now to establish the discrete Gagliardo-Nirenberg inequality corresponding to the discrete 2D counterpart of (3):

Theorem 3.7. Let Ω "sx, xrˆsy, yr be an open set of R 2 , v P H M pΩq, 1 ď p, r ă 8 and 1 ď q ď 8 such that 2 p " 1 r `1 q .
Then, we have:

}∇ h v} L p pΩq ď 2 ´1 2q C GN }∇ 2 h v} 1{2 L r pΩq }v} 1{2 L q pΩq , (28)
where C GN is the constant arising in Theorem 3.1.

Proof. First of all, from v P H M pΩq we define some discrete one-variable functions: For j P v1, M w, v pjq P H M pI x q with I x "sx, xr and

}δ x v pjq } p L p pIxq " ż Ix |δ x v pjq | p dx.
For i P v1, N w, v piq P H M pI y q with I y "sy, yr and

}δ y v piq } p L p pIyq " ż Iy |δ y v piq | p dy.
We have:

}∇ h v} p L p pΩq " ż Ω |δ x v| p dx `żΩ |δ y v| p dx " M ÿ j"1 k j }δ x v pjq } p L p pIxq `N ÿ i"1 h i }δ y v piq } p L p pIyq .
From Theorem 3.1 and Hölder inequality, we get:

}∇ h v} p L p pΩq ď C p GN ˜M ÿ j"1 k j }δ xx v pjq } p 2 L r pIxq }v pjq } p 2 L q pIxq `N ÿ i"1 h i }δ yy v piq } p 2 L r pIyq }v piq } p 2 L q pIyq ḑ C p GN ´}δ xx v} p 2 L r pΩq `}δ yy v} p 2 L r pΩq ¯}v} p 2 L q pΩq ď 2 p 2r ´1 C p GN }∇ 2 h v} p 2 L r pΩq }v} p 2
L q pΩq , so that (28) holds.

Remark 3.8. In the following section, we will focus on the case p " 4, r " 2, q " 8:

}∇ h v} L 4 pΩq ď C GN }∇ 2 h v} 1{2 L 2 pΩq }v h } 1{2 L 8 pΩq . ( 29
)
Remark 3.9. Up to a slight modification of the prefactor 2 ´1 2q in 2 ´1 q , the discrete Gagliardo-Nirenberg inequality (28) also holds when the domain Ω is a subset of R 3 . Moreover, the equality (86) also holds true in the three-dimensional context. Indeed, the definitions of the discrete operators, scalar product and norms can be done in a similar way to the two-dimensional case. An induction argument is used to conclude the proof of Theorem 3.7, and a term-by-term identification can be done to obtain (86).

4 Finite Volume scheme and a priori estimates

Model and continuous results

In this section, we are interested in a convection-diffusion equation involving a Joule effect term, given by:

B t u `∇ ¨pu vq `λ |∇u| 2 ´λu ∆u " 0, @ x P Ω, @ t Ps0, T s, (30a)
∇upx, tq ¨n " 0, @ x P BΩ, @ t Ps0, T s, (30b)

upx, 0q " u 0 pxq, in Ω, (30c)
where Ω "sx, xrˆsy, yrĂ R 2 , n is the outward unit normal vector to BΩ, T ą 0 is an arbitrary finite time horizon, λ ą 0 is a fixed parameter, and the vector field v : Q T " Ωˆr0, T s Ñ R 2 is divergence free and satisfies a no-slip boundary condition, i.e. vpx, tq " 0 for all x P BΩ and t P r0, T s. The system (30) can be seen as a particular case of a global low-Mach model with temperature dependent viscosity, in the case where v is a given datum of the problem (see e.g. [START_REF] Huang | On the strong solution of the ghost effect system[END_REF][START_REF] Calgaro | Approximation by an iterative method of a low-Mach model with temperature dependent viscosity[END_REF]). A local wellposedness result for strong solutions to (30) has been established in [START_REF] Calgaro | Approximation by an iterative method of a low-Mach model with temperature dependent viscosity[END_REF]Theorem 1]. More precisely, assuming that u 0 P H 2 N pΩq " tw P H 2 pΩq s.t. ∇wpxq ¨npxq " 0 for a.e. x P BΩu, that the convective velocity satisfies v P L 2 p0, T ; V 0 pΩq X pH 2 pΩqq 2 q with V 0 pΩq " tw P pH 1 0 pΩqq 2 ; ∇ ¨w " 0, in Ωu, (31) and if there exist two real numbers u 5 and u 7 such that 0 ă u 5 ď u 0 pxq ď u 7 , @x P Ω, (

then there exists T ą 0 such that the system (30) admits a unique solution with the following regularity:

u P L 2 p0, T ; H 3 pΩqq X L 8 p0, T ; H 2 N pΩqq, B t u P L 2 p0, T ; H 1 pΩqq (33) with 0 ă u 5 ď u ď u 7 a.e. Q T . (34)
In this paper, we rather work with a weaker notion of solutions demanding for less regularity than (33). Definition 4.1. A function u is said to be a global in time weak solution to Problem (30) if u P L 8 pQ T ; ru 5 , u 7 sq X L 8 pp0, T q; H 1 pΩqq with B t u and ∇ 2 u P L 2 pQ T q, if ∇u ¨n " 0 on BΩ ˆp0, T q, and if (30a) holds (with each term belonging to L 2 pQ T q). With such a lower regularity requirement, we are able to prove the existence of a global-in-time weak solution.

Theorem 4.2. Suppose u 0 P H 1 pΩq and that the assumptions (31)-(32) are satisfied. If

u 7 ´u5 ď δ, (35)
for some δ ą 0 small enough (with a condition similar to the one of Theorem 4.4 below), then there exists a weak solution u satisfying }u ´u} L 8 pΩˆR`q ď δ, with u " u 5 `u7 2 ą 0. Moreover there exists C ě 0 such that for all t ą 0:

}uptq ´u} 2 H 1 pΩq `ż t 0 ´}∇upsq} 2 L 2 pΩq `}∆upsq} 2 L 2 pΩq ¯ds ď C ˆ}u 0 ´u} 2 H 1 pΩq `ż t 0 }∇vpsq} 2 L 2 pΩq ds ˙. ( 36
)
The existence of such a global in time weak solution is a by-product of the Theorem 4.4 on the convergence of the finite volume scheme to be introduced in the next section. Note also that the assumption (35) is necessary to prove that the system (30) admits a unique global-in-time strong solution (33) with (34) (see [START_REF] Huang | On the strong solution of the ghost effect system[END_REF]).

The Finite Volume scheme

We notice that ∇ ¨pu∇uq " |∇u| 2 `u∆u. Then, the way to discretize the Joule effect term |∇u| 2 arising in (30a) must be consistent with the non-linear diffusion one. This is important in order to ensure some properties on the numerical solution, such as some maximum principles which hold at the continuous level. Moreover, the non-conservative way to write the diffusion term is consistent with the analysis that we will do, which mimics the continuous one. A rather similar Finite Volume (FV) scheme was initially introduced in [START_REF] Calgaro | A combined finite volume-finite element scheme for a low-Mach system involving a Joule term[END_REF].

In addition to the notations of subsection 2.2.1, we denote E " E H Y E V the set of the horizontal and vertical edges of the mesh, i.e.

E H " ! σ i,j`1 2 "sx i´1 2 , x i`1 2 rˆty j`1 2 u, i P v1, N w, j P v0, M w ) , E V " ! σ i`1 2 ,j " tx i`1 2 uˆsy j´1 2 , y j`1 2 r, i P v0, N w, j P v1, M w
) .

Now we introduce the definition of a uniform mesh in each direction.

Definition 4.3. A mesh M is said uniform in each direction if h i " h x for i P v1, N w and k j " h y for j P v1, M w.

From now on and for the sake of simplicity, we assume that the mesh M is uniform in each direction. As usual in the finite volume context, the size of the mesh is then defined as the diameter of the cells, i.e.

h " b h 2 x `h2 y .
We also introduce the transmissibility coefficient, given by

a σ " h y h x for σ P E V and a σ " h x h y for σ P E H .
Let us introduce the space

V E,0 pΩq " v h " pv 1,h , v 2,h q P H 0 M pΩq | div h v h " 0 ( , (37)
where the operator div h is defined from

H M pΩq in H M pΩq by div h v h " δ x v 1,h `δẙ v 2,h .
Let E i,j be the boundary of the control volume C i,j (i P v1, N w, j P v1, M w). For σ P E i,j , we denote by n i,j,σ the exterior unit normal vector to σ. Given a fixed but arbitrary finite time horizon T ą 0, we split the time interval r0, T s in a uniform partition of time step τ " T {N T for some N T P N ą0 , we define t n " nτ (0 ď n ď N T ) (so that r0, T s " Ť 0ďnăN T rt n , t n`1 s). For any velocity field v " pv 1 , v 2 q P L 2 p0, T ; V 0 pΩq), we define v hτ " pv 1,hτ , v 2,hτ q P L 2 p0, T ; V E,0 pΩqq by setting v 1,hτ px, tq " v n i`1{2,j if px, tq P C i`1{2,j ˆpt n , t n`1 q, (38a)

v 2,hτ px, tq " v n i,j`1{2 if px, tq P C i,j`1{2 ˆpt n , t n`1 q, (38b)
with

v n i`1 2 ,j " 1 τ ż t n`1 t n 1 h y ż σ i`1 2 ,j
v 1 px, sq dσpxq ds, i P v1, N ´1w, j P v1, M w, (38c)

v n i,j`1 2 " 1 τ ż t n`1 t n 1 h x ż σ i,j`1 2
v 2 px, sq dσpxq ds, i P v1, N w, j P v1, M ´1w. (38d)

The integrands in the above formulas have to be understood as the traces of pv 1 , v 2 q P L 2 p0, T ; pH 2 pΩqq 2 q on the edges. Moreover, since H 2 pΩq embeds in L 8 pΩq (this also holds true in the three-dimensional setting), then

|v n i,j`1 2 | ď 1 τ ż t n`1 t n }vp¨, tq} 8 dt
We define moreover v n h P V E,0 pΩq by:

v n h pxq " pv n 1,h pxq, v n 2,h pxqq " 1 τ ż t n`1
t n v hτ px, sq ds @ x P Ω.

We infer from Jensen's inequality that

}v hτ } L 2 p0,T ;L 8 pΩqq ď }v} L 2 p0,T ;L 8 pΩqq ď C Ω }v} L 2 p0,T :H 2 pΩqq ( 39
)
with C Ω being the continuity constant for the injection of H 2 pΩq into L 8 pΩq.

The initial data u 0 is discretized into

u 0 i,j " 1 h x h y ż Ci,j u 0 pxqdx, i P v1, N w, j P v1, M w. (40)
Assuming that u n h P H M pΩq is a known approximation of up¨, t n q, we are looking for an approximation u n`1 h P H M pΩq of up¨, t n`1 q, with

u n h pxq " u n i,j if x P C i,j , n ě 0.
The space-time approximate solution u hτ P L 8 p0, T ; H M pΩqq is then defined almost everywhere by u hτ px, tq " u n`1 h pxq if t P pt n , t n`1 s.

The scheme is obtained by integrating (30a) on each C i,j P M, leading to

h x h y u n`1 i,j ´un i,j τ `ÿ σPEi,j v n i,j,σ u n`1 i,j,σ,``λ h x h y `Ji,j pu n`1 h q ´un`1 i,j p∆ h u h q n`1 i,j ˘" 0. ( 41
)
In the above equation (41), we defined v n i,j,σ by

v n i,j,σ " $ & % ˘hy v n i˘1 2 ,j if σ " σ i˘1 2 ,j P E V , ˘hx v n i,j˘1 2 if σ " σ i,j˘1 2 P E H ,
and by u n`1 i,j,σ,`t he upstream choice for the convection term defined for σ P E i,j :

u n`1 i,j,σ,`" # u n`1 i,j if v n i,j,σ ě 0, u n`1 i,j,σ otherwise, with u n`1 i,j,σ " $ ' ' & ' ' % u n`1 i˘1,j if σ " σ i˘1 2 ,j P E V , u n`1 i,j˘1 if σ " σ i,j˘1 2 P E H , u n`1 i,j if σ Ă BΩ.
The discretization of the Joule effect term is more original as we set

J i,j pu n`1 h q " 1 h x h y ÿ σPEi,j a σ `pu n`1 i,j ´un`1 i,j,σ q `˘2 , (42)
where a `" maxp0, aq. We also denote by

J h pu n`1 h qpxq " J i,j pu n`1 h q if x P C i,j . (43)
This discretization of the Joule effect term |∇u| 2 can be thought as some dual counterpart of the upstream convection, see [START_REF] Cancès | A variational finite volume scheme for Wasserstein gradient flows[END_REF]. It enjoys the following key property for the preservation of the maximum principle:

u n`1 i,j ď u n`1 i,j,σ for all σ P E i,j ùñ J i,j pu n`1 h q " 0, transposing to the discrete setting the fact that |∇u| 2 vanishes at the minima of u.

Besides the second order discrete Gagliardo-Nirenberg inequality stated in Section 3, the main result of the paper can be gathered in the following statement.

Theorem 4.4. Let u 0 P H 1 pΩq be such that u 5 ď u 0 ď u 7 for some (strictly) positive constants u 5 , u 7 , then the numerical scheme (40)-( 41)-(42) admits a unique iterated in time solution u hτ with u 5 ď u hτ ď u 7 a.e. in Q T . Moreover, if u 7 ´u5 ă δ with 0 ă δ ă 2 pC GN q 2 p a 1 `u5 2 ´1q, then, up to a subsequence,

u hτ ÝÑ h,τ Ñ0 u a.e. in Q T
where u is a weak solution to the continuous problem in the sense of Definition 4.1.

Remark 4.5. The constraint on δ might look restrictive but it is imposed by the global well-posedness of the continuous problem. We emphasize that the proposed convergence result applies to (30), but it is also motivated by a practical application on a ghost effect system (see [START_REF] Levermore | Local well-posedness of a ghost system effect[END_REF][START_REF] Huang | On the strong solution of the ghost effect system[END_REF]). Ghost effect systems are formally derived to describe regimes in which the compressible Navier-Stokes system is incomplete, in particular when the classical heat-conduction equation fails to correctly describe the temperature field of the gas. In such a physical context, the parameter δ is expected to be small. The analysis done in this work can be considered as part of the analysis of a numerical scheme for a ghost effect system or a low Mach model expressed in velocity, pressure and temperature variables, as proposed in [START_REF] Calgaro | A combined finite volume-finite element scheme for a low-Mach system involving a Joule term[END_REF] where some numerical tests are also presented.

The two next sections are devoted to the proof of Theorem 4.4. Moreover, finer convergence properties will be derived along the proof, especially in Section 6.1.

Numerical analysis at fixed grid

The goal of this section is to prove the well-posedness of the numerical scheme as well as estimates which are uniform with respect to the grid. Those estimates will serve as cornerstones for the convergence proof reported in Section 6.

Maximum principle and existence of a discrete solution

We first establish a uniform L 8 a priori estimate on the discrete solution, on which we will rely to show the existence of a discrete solution u h to the scheme (41).

Proposition 5.1. Assume that there exist two positive constants u 5 , u 7 such that

0 ă u 5 ď u 0 ď u 7 . ( 44
)
Then for all n P v1, N T w the finite volume scheme (40)-( 41) admits a unique solution u n h P H M pΩq which satisfies 0 ă u 5 ď u n i,j ď u 7 @ i P v1, N w, j P v1, M w, n P v1, N T w.

Proof. The proof is done by induction over n. The initialization for n " 0 is straightforward in view of (44) and the definition (40) of the initial discrete solution. We perform a harmless modification of the scheme, which now writes

h x h y u n`1 i,j ´un i,j τ `ÿ σPEi,j v n i,j,σ u n`1
i,j,σ,λ h x h y `Ji,j pu n`1 h q ´pu n`1 i,j q `p∆ h u h q n`1 i,j ˘" 0, (46) instead of (41). Of course, once (45) is established, we get that solutions to (46) are also solutions to (41). The modified scheme (46) can be rewritten in the compact form

F i,j pu n`1 h
q " u n i,j , i P v1, N w, j P v1, M w, where F h " pF i,j q i,j : H M pΩq Ñ H M pΩq is increasing w.r.t. u n`1 i,j and non-increasing w.r.t. u n`1 k, as soon as pk, q ‰ pi, jq.Moreover, since v n h is discrete divergence free, one has for all κ P R that F i,j pκ h q " κ, i P v1, N w, j P v1, M w,

where κ h is the element of H M pΩq which is constant equal to κ. The Jacobian matrix

Jpu n`1 h q "
´BFi,j Bu k, pu n`1 h q ¯pi,jq,pk, q is a M -matrix in the sense of [START_REF] Hackbusch | Elliptic Differential Equations: Theory and Numerical Treatment[END_REF]Definition 4.8].

Let ǔn`1 h be another solution to (46) corresponding to some previous step value ǔn h , then

F h pu n`1 h q ´Fh pǔ n`1 h q " Jpu n`1 h , ǔn`1 h qpu n`1 h ´ǔ n`1 h q " u n h ´ǔ n h , where Jpu n`1 h , ǔn`1 h q " ż 1 0 Jpǔ n`1 h `tpu n`1 h ´ǔ n`1 h qqdt is also a M matrix. It is in particular invertible with Jpu n`1 h , ǔn`1 h q ´1 ě 0 component- wise. Therefore, u n h ě ǔn h ùñ u n`1 h ě ǔn`1 h .
This yields in particular the uniqueness of the solution to (46), as well as the maximum principle (45) thanks to (47) if one chooses ǔn h " ǔn`1 h constant equal to u 5 or u 7 . Finally, the existence of a solution to the modified scheme (46), and thus to the original one (41) is obtained thanks to some classical topological degree argument. We refer to [START_REF] Leray | Topologie et équations fonctionnelles[END_REF][START_REF] Deimling | Nonlinear functional analysis[END_REF] for a general presentation of the topological gradient theory, and to [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF] for its first (up to our knowledge) use in the context of finite volumes.

Further estimates

The goal of this section is to establish the next estimates required to establish the convergence of the scheme. The main and next one is a L 8 loc pH 1 q X L 2 loc pH 2 N q estimate obtained under some smallness assumption on the data. Proposition 5.2. Assume that (44) holds. Then, there exists δ ą 0 such that if

u 7 ´u5 ď δ, ( 48
)
then there exists c ą 0 only depending on Ω, u 0 , v, λ, δ and T such that the solution u n h P H M pΩq of the scheme (41) built at Proposition 5.1 satisfies the following estimates:

}∇ h u n h } L 2 pΩq ď c, @n P v1, N T w, (49)
N T ´1 ÿ n"0 τ ´}∆ h u n`1 h } 2 L 2 pΩq `}J h pu n`1 h q} 2 L 2 pΩq ¯ď c. ( 50
)
Proof. Before addressing the properties of u n h , n ě 1, induced by the scheme, let us first remark that

}∇ h u 0 h } L 2 pΩq ď }∇u 0 } L 2 pΩq ( 51
)
thanks to the definition (40) of u 0 h and to successive uses of Jensen's inequality and Fubini's theorem. We refer for instance to [START_REF] Eymard | Finite volume methods[END_REF]Lemma 9.4] for an extension of (51) to the more complex case of non-structured grids.

Given n P v0, N T ´1w, we multiply (41) by p´∆ h u h q n`1 i,j and we sum for i " 1, . . . , N and j " 1, . . . , M :

´żΩ u n`1 h ´un h τ ∆ h u n`1 h dx `λ ż Ω u n`1 h p∆ h u n`1 h q 2 dx " ÿ Ci,j PM ÿ σPEi,j v n i,j,σ u n`1 i,j,σ,`p ∆ h u h q n`1 i,j `λ ż Ω J pu n`1 h q∆ h u n`1 h dx. ( 52
)
Owing to (85) and to the convexity inequality pb ´aqb ě b 2 2 ´a2 2 , the first term in the left-hand side can be underestimated by

´żΩ u n`1 h ´un h τ ∆ h u n`1 h ě 1 2τ }∇ h u n`1 h } 2 L 2 pΩq ´1 2τ }∇ h u n h } 2 L 2 pΩq . ( 53
)
For the second term of the left-hand side, the maximum principle (see Proposition 5.1) implies that

λ ż Ω u n`1 h p∆ h u n`1 h q 2 dx ě λ u 5 }∆ h u n`1 h } 2 L 2 pΩq . ( 54
)
For the convection term, we have two contributions corresponding respectively to the centered approximation for the convection and to the numerical diffusion stemming from the upwinding, see (98). We recall that the properties of the discrete interpolation operators are collected in Appendix A. Concerning the centered part, we deduce from (97), (99) and Lemma B.1 that

ÿ Ci,j PM ÿ σPEi,j v n i,j,σ u n`1 i,j,σ,c p∆ h u h q n`1 i,j " ´T p1,2q 1 ´T p2,2q 1 ´T p1,2q 2 ´T p2,2q 2
, ( 55
)
with:

T p1,2q 1 " ż Ω π x pδ x v n 1,h ¨πx pδ x u n`1 h qq ¨δx u n`1 h dx, T p2,2q 1 " ż Ω π ẙ pδ x v n 2,h ¨πx pδ y u n`1 h qq ¨δx u n`1 h dx, T p1,2q 2 " ż Ω π x pδ y v n 1,h ¨πy pδ x u n`1 h qq ¨δy u n`1 h dx,
and

T p2,2q 2 " ż Ω π y pδ ẙ v n 2,h ¨πẙ pδ y u n`1 h qq ¨δy u n`1 h dx.
The combination of Proposition A.5, standard Hölder inequalities together with Proposition A.6 yields

|T p1,2q 1
| ď }δ x v n 1,h } L 2 pΩq }π x pδ x u n`1 h q} 2 L 4 pΩq ď }δ x v n 1,h } L 2 pΩq }δ x u n`1 h } 2 L 4 pΩq , |T p2,2q 1
| ď }δ x v n 2,h } L 2 pΩq }π x pδ y u n`1 h q} L 4 pΩq }π y pδ x u n`1 h q} L 4 pΩq ď }δ x v n 2,h } L 2 pΩq }δ y u n`1 h } L 4 pΩq }δ x u n`1 h } L 4 pΩq , |T p1,2q 2
| ď }δ y v n 1,h } L 2 pΩq }π y pδ x u n`1 h q} L 4 pΩq }π x pδ y u n`1 h q} L 4 pΩq ď }δ y v n 1,h } L 2 pΩq }δ x u n`1 h } L 4 pΩq }δ y u n`1 h } L 4 pΩq , |T p2,2q 2
| ď }δ ẙ v n 2,h } L 2 pΩq }π ẙ pδ y u n`1 h q} 2 L 4 pΩq ď }δ ẙ v n 2,h } L 2 pΩq }δ y u n`1 h } 2 L 4 pΩq , whence the estimate ÿ Ci,j PM ÿ σPEi,j v n i,j,σ u n`1 i,j,σ,c p∆ h u h q n`1 i,j ď 1 2 λ }∇ h v n h } 2 L 2 pΩq `λ 2 ´}δ x u n`1 h } 2 L 4 pΩq `}δ y u n`1 h } 2 L 4 pΩq ¯2 ď 1 2 λ }∇ h v n h } 2 L 2 pΩq `λ pC GN q 4 pu 7 ´u5 q 2 8 }∆ h u n`1 h } 2 L 2 pΩq . ( 56
)
For the last inequality we used the Gagliardo-Nirenberg inequality (29) applied to

u n`1 h ´u5 ` u7
2 P H M pΩq) combined with identity (86). Let us now focus on the numerical diffusion part corresponding to the second term in (98). Since

ˇˇu n`1 i,j,σ,`´u n`1 i,j,σ,c ˇˇ" 1 2 |u n`1 i,j
´un`1 i,j,σ |, @ σ P E i,j , one can rewrite

A :" ˇˇhxhy ÿ Ci,j PM ÿ σPEi,j v n i,j,σ `un`1 i,j,σ,`´u n`1 i,j,σ,c ˘p∆ h u h q n`1 i,j ˇď h x h y 2 ÿ Ci,j PM ˇˇp∆ h u h q n`1 i,j ˇˇÿ σPEi,j |v n i,j,σ ||u n`1 i,j ´un`1 i,j,σ |
which together with Young's inequality leads to

A ď B ` 4 }∆ h u n`1 h } 2 L 2 pΩq , with B " 1 4 h x h y ÿ Ci,j PM ¨ÿ σPEi,j |v n i,j,σ ||u n`1 i,j ´un`1 i,j,σ | '2 ,
and where ą 0 will be fixed later on. Using now the elementary pa `b `c `dq 2 ď 4pa 2 `b2 `c2 `d2 q and Young's inequality, we get that

B ď 1 h x h y ÿ Ci,j PM ÿ σPEi,j |v n i,j,σ | 2 |u n`1 i,j ´un`1 i,j,σ | 2 ď 1 4α h x h y ÿ Ci,j PM ÿ σPEi,j |v n i,j,σ | 4 `α h x h y ÿ Ci,j PM ÿ σPEi,j |u n`1 i,j ´un`1 i,j,σ | 4 ď 1 2α }v n h } 4 L 4 pΩq `2α h 4 }∇ h u n`1 h } 4 L 4 pΩq ď pC S q 4 2α }∇ h v n h } 4 L 2 pΩq `α 2 pC GN q 4 h 4 pu 7 ´u5 q 2 }∆ h u n`1 h } 2 L 2 pΩq ,
where the last inequality is a consequence of the discrete Sobolev inequality (see for instance [START_REF] Eymard | Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes sushi: a scheme using stabilization and hybrid interfaces[END_REF])

}v n h } L 4 pΩq ď C S }∇ h v n h } L 2 pΩq
and of the discrete Gagliardo-Nirenberg inequality (29) combined with identity (86).

For the parameter α ą 0, we choose α " λ {p4h 4 q, so that

B ď 2h 4 pC S q 4 λ 2 }∇ h v n h } 4 L 2 pΩq `λ 8 pC GN q 4 pu 7 ´u5 q 2 }∆ h u n`1 h } 2 L 2 pΩq .
Setting " λpC GN q 4 pu 7 ´u5 q 2 , one gets that

A ď λ ˆhC S λpC GN q 2 pu 7 ´u5 q ˙4 }∇ h v n h } 4 L 2 pΩq `3λ 8 pC GN q 4 pu 7 ´u5 q 2 }∆ h u n`1 h } 2 L 2 pΩq .
(57) Finally, applying Cauchy-Schwartz inequality to the last term in (52) leads to

λ ż Ω J h pu n`1 h q∆ h u n`1 h dx ď λ}∆ h u n`1 h } L 2 pΩq }J h pu n`1 h q} L 2 pΩq .
The definition (42) and (43) of J h is so that

}J h pu n`1 h q} 2 L 2 pΩq " 1 h x h y ÿ Ci,j PM ¨ÿ σPEi,j a σ `pu n`1 i,j ´un`1 i,j,σ q `˘2 '2 ď 4 h x h y ÿ Ci,j PM ÿ σPEi,j a 2 σ `pu n`1 i,j ´un`1 i,j,σ q `˘4 ď 4h x h y ÿ Ci,j PM ÿ σPEi,j ˜pu n`1 i,j ´un`1 i,j,σ q hσ ¸4
, with h σ " h x if σ P E V and h σ " h y if σ P E H is the distance between the cell centers |x i,j ´xi,j,σ |. Due to the positive part, each edge σ is counted once in the last sum, and we deduce that

}J h pu n`1 h q} L 2 pΩq ď 2 ¨hx h y ÿ σPE ˜un`1 i,j ´un`1 i,j,σ h σ ¸4' 1{2 " 2}∇ h u n`1 h } 2 L 4 pΩq .
Applying again (29) and (86), one gets that

}J h pu n`1 h q} L 2 pΩq ď 2pC GN q 2 pu 7 ´u5 q}∆ h u n`1 h } L 2 pΩq , (58)
and then that

λ ż Ω J h pu n`1 h q∆ h u n`1 h dx ď 2λpC GN q 2 pu 7 ´u5 q}∆ h u n`1 h } 2 L 2 pΩq . (59)
Eventually, combining (53)-( 59) in (52) leads to

1 2τ ´}∇ h u n`1 h } 2 L 2 pΩq ´}∇ h u n h } 2 L 2 pΩq λ}∆ h u n`1 h } 2 L 2 pΩq ˆu5 ´1 2 pC GN q 4 δ 2 ´2pC GN q 2 δ ˙ď ˜1 2λ `C1 ˆh δ ˙4¸} ∇ h v n h } 4 L 2 pΩq
with C 1 depending only on Ω (via C GN and C S ) and on λ. For δ ă

2 pC GN q 2 p a 1 `u5 2
1q, the term in front of }∆ h u n`1 h } 2 is positive. Owing to Lemma B.2, the above righthand side is bounded by some quantity not depending on the mesh.

From the above estimate, we deduce a uniform L 2 pQ T q estimate on δ τ u hτ P L 8 p0, T ; H M q defined by δ τ u hτ p¨, tq "

u n`1 h ´un h τ if t P rt n , t n`1 q. ( 60
)
Then the following estimate directly follows from the use of the estimates of Proposition 5.2 in the scheme (41).

Corollary 5.3. Under the assumptions of Proposition 5.2, there exists C ě 0 depending only on Ω, u 0 , v, λ, δ and T such that

ij Q T |δ τ u hτ | 2 dxdt ď C.

Convergence of the finite volume scheme

The purpose of this section is to establish the convergence of the scheme thanks to compactness arguments. Given pM m q mě0 a sequence of admissible meshes with size h m tending to 0 as m tends to `8, and given pτ m q mě0 be a sequence of positive time steps tending to 0, then denoting by pu hmτm q mě0 the corresponding sequence of approximate solution provided by Proposition 5.1, then one aims to show that, up to the extraction of a subsequence, u hmτm tends to a weak solution u to (30) in the sense of Definition 4.1. Our proof is based on compactness arguments. We start in Section 6.1 to establish some compactness properties on the approximate solutions pu hmτm q m , then the limit value will be identified as a weak solution to the problem in Section 6.2.

Some compactness properties

First, it follows from Proposition 5.1 that there exists u P L 8 pQ T q with u 5 ď u ď u 7 such that, up to a subsequence, there holds

u hmτm ÝÑ mÑ`8 u in the L 8 pQ T q-weak-‹ sense. (61)
Moreover, thanks to the (uniform w.r.t. m) L 2 pQ T q bounds on ∇ hm u hmτm and δ τm u hmτm respectively established in Proposition 5.2 and Corollary 5.3, we can mimic the technics detailed in [START_REF] Eymard | Finite volume methods[END_REF] to get estimates on the space-and-time translates

ż T ´ζ 0 ż Ω ξ |u hmτm px`ξ, t`ζq´u hmτm px, tq| 2 dxdt ď C `ζ2 `|ξ| 2 ˘, ζ P p0, T q, ξ P R 2 ,
with C not depending on m, and with Ω ξ " tx P Ω | x `ξ P Ωu. This in particular yields the relative compactness of the sequence pu hmτm q m in L 2 pQ T q thanks to Kolmogorov's compactness criterion. Therefore, up to the extraction of a subsequence, we get that

u hmτm ÝÑ mÑ`8 u a.e. in Q T . (62)
Besides, we deduce from Estimate (50) that, still up to a subsequence, there holds

∆ hm u hmτm ÝÑ mÑ`8
∆u weakly in L 2 pQ T q.

(63) Indeed, the (uniform w.r.t. m) L 2 pQ T q bound on ∆ hm u hmτm ensures the existence of some weak limit d P L 2 pQ T q. Then following the program of [START_REF] Eymard | Finite volume methods[END_REF], the identification of d " ∆u is then obtained in the distributional sense. Similarly, we deduce from Corollary 5.3 that

δ τm u hmτm ÝÑ mÑ`8 B t u weakly in L 2 pQ T q. ( 64
)
Concerning the sequence p∇ hm u hmτm q m , we have the uniform L 8 p0, T ; L 2 pΩqq 2 estimate (49) as well as a L 4 pQ T q 2 estimate stemming from the combination of ( 29), ( 45) and (50). After identifying the weak limit in the distributional sense once again, one gets that

∇ hm u hmτm ÝÑ mÑ`8
∇u in the L 4 pQ T q-weak and L 8 p0, T ; L 2 pΩqq 2 -weak-‹ senses.

(65) Further compactness is required to pass in the limit in the Joule effect term J hm pu hmτm q, whence next lemma. Lemma 6.1. Up to extraction of a subsequence, the following convergence holds:

∇ hm u hmτm ÝÑ mÑ`8 ∇u a.e. in Q T .
Proof. The proof relies on some discrete Aubin-Lions-Simon lemma. In the proof, we make use of the result presented in [START_REF] Andreianov | A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs[END_REF] but we stress that a proof building on [START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic PDEs-application to a turbulence model[END_REF][START_REF] Gallouët | Discrete functional analysis tools for some evolution equations[END_REF] is also possible.

We proceed direction-wise, proving that δ x u hmτm P p H 0 Mm pΩq converges pointwise towards B x u. Of course, proving the convergence of δ y u hmτm P r H 0 Mm pΩq towards B y u is similar.

The combination of Estimate (50) with identity (86) provides that

}∇ hm δ x u hmτm } L 2 pQ T q 2 ď C
for some C not depending on m, providing some compactness with respect to the space variable on pδ x u hmτm q m . On the other hand, given ϕ P C 8 c pQ T q, and denoting by r ϕ hmτm the piecewise constant in time and space function defined by

r ϕ hmτm px, tq " 1 τ m h x,m h y,m ż pn`1qτm nτm ż C i`1{2,j ϕ dxdt, then ij Q T δ x δ τm u hmτm ϕ dxdt " ij Q T δ x δ τm u hmτm r ϕ hmτm dxdt " ´ij Q T δ τm u hmτm δ x r ϕ hmτm dxdt.
Applying Cauchy-Schwarz inequality and using Corollary 5.3 and [12,Lemma 9.4] yields

ij Q T δ x δ τm u hmτm ϕ dxdt ď C}∇ hm r ϕ hmτm } L 2 pQ T q 2 ď C}∇ϕ} L 2 pQ T q 2 .
We can thus apply Theorem 3.9 of [START_REF] Andreianov | A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs[END_REF] which ensures that δ x u hmτm converges pointwise.

Because of (65), the limit is B x u.

Notice that thanks to (65) and Lemma 6.1, we can apply Vitali's convergence theorem and claim that

∇ hm u hmτm ÝÑ mÑ`8 ∇u in L 2 pQ T q. ( 66
)
With the above compactness properties, we have enough material to to the limit in the scheme. This is the purpose of next section.

Limits are weak solutions

As a preliminary to the identification of any limit value u of u hmτm as in Section 6.1 as a weak solution, let us first show the consistency of the discretization (42) and (43) of the Joule term.

Lemma 6.2. Up to a subsequence there holds

J hm pu hmτm q ÝÑ mÑ`8
|∇u| 2 weakly in L 2 pQ T q.

Proof. First, it follows from (66) that

|∇ hm u hmτm | 2 ÝÑ mÑ`8 |∇u| 2 in L 1 pQ T q. ( 67
)
On the other hand, we deduce from (50) that there exists some J P L 2 pQ T q such that J hm pu hmτm q ÝÑ mÑ`8

J weakly in L 2 pQ T q.

Let us now identify J as |∇u| 2 in the distributional sense. Let ϕ P C 8 c pQ T q, and define ϕ hmτm by setting ϕ n i,j " ϕpx i,j , t n q for all C i,j P M m and all n P v1, N T,m w, then ij

Q T `Jhm pu hmτm q ´|∇ hm u hmτm | 2 ˘ϕ dxdt ď R m pϕq `Sm pϕq, (68)
´un i,j ˘2 ď Ch m ÝÑ mÑ`8 0, (70)
the last inequality being a consequence of (49). Then we deduce from (67)-(70) that J " |∇u| 2 , concluding the proof of Lemma 6.2.

Our next lemma is about the boundary conditions for the limit u, which is shown to belong to L 2 p0, T ; H 2 N pΩqq. Lemma 6.3. Let u be a limit of pu hmτm q m as in Section 6.1, then ∇u ¨n " 0 on BΩ ˆp0, T q.

Proof. First note that since ∆u belongs to L 2 pQ T q, cf. (63) and since Ω is convex, then u belongs to L 2 p0, T ; H 2 pΩqq and ∇u P L 2 p0, T ; H 1 pΩqq 2 admits strong traces in L 2 p0, T ; H 1{2 pBΩqq 2 Ă L 2 pBΩ ˆp0, T qq 2 . Let us show that ´Bx u " ∇u ¨n " 0 on `txu ˆpy, yq ˘ˆp0, T q, the treatment of the other parts of the boundary being similar.

We proceed as in Since v n h is divergence free, one has ř σPEi,j v n i,j,σ " 0, so that W n`1 i,j rewrites

W n`1 i,j " 1 h x,m h y,m ÿ σPEi,j v n i,j,σ pu n`1 i,j,σ,`´u n`1 i,j q.
Then it follows from the definition of u n`1 i,j,σ,`t hat v n i,j,σ pu n`1 i,j,σ,`´u n`1 i,j q " pv n i,j,σ q ´pu n`1 i,j

´un`1 i,j,σ q, where pv n i,j,σ q ´" maxp0, ´vn i,j,σ q denotes the negative part of v n i,j,σ .

ij Q T |W hmτm | 2 dxdt " N T ,m ÿ n"1 τ m ÿ Ci,j PMm 1 h x,m h y,m ¨ÿ σPEi,j pv n´1 i,j,σ q ´pu n i,j ´un i,j,σ q '2 ď 4 N T ,m ÿ n"1 τ m ÿ Ci,j PMm 1 h x,m h y,m ÿ σPEi,j `pv n´1 i,j,σ q ´˘2 `pu n i,j ´un i,j,σ q ˘2 .
Bearing in mind the definition of v n i,j,σ , we get that

ij Q T |W hmτm | 2 dxdt ď 2 N T ,m ÿ n"1 τ m }v n h } 2 8 ÿ Ci,j PMm ÿ σPEi,j a σ pu n i,j ´un i,j,σ q 2 .
Making use of ( 49) and (39), one eventually gets that

ij Q T |W hmτm | 2 dxdt ď C.
In particular, there exists some W ˚P L 2 pQ T q such that

W hmτm ÝÑ mÑ`8 W ˚weakly in L 2 pQ T q. (79)
Let us identify W ˚as v ¨∇u in the distributional sense. Let ϕ P C 8 c pQ T q, then defining ϕ hmτm as the piecewise constant in time and space function built from the cell values ϕ n i,j " ϕpx i,j , t n q, then ˇˇˇˇˇij

Q T W hmτm pϕ ´ϕhmτm q dxdt ˇˇˇˇˇď ij Q T |W hmτm | |ϕ ´ϕhmτm |dxdt. (80)
Due to the regularity of ϕ and the boundedness in L 2 pQ T q of W hmτm , the above right-hand side tends to 0 as m tends to `8. We write

ij Q T W hmτm ϕ hmτm dxdt " T p1q m pϕq `T p2q m pϕq with T p1q m pϕq " 1 2 N T ,m ÿ n"1 τ m ÿ Ci,j PMm ϕ n i,j ÿ σPEi,j v n´1 i,j,σ `un i,j,σ ´un i,j " ij Q T v hmτm ¨∇hm u hmτm ϕ hmτm dxdt, and
T p2q m pϕq " 1 2 N T ,m ÿ n"1 τ m ÿ σPEm v n´1 i,j,σ `un i,j ´un i,j,σ ˘`ϕ n i,j ´ϕn i,j,σ ˘.
In particular, by similar arguments to those which lead to (79) combined with the regularity of ϕ, one gets that

ˇˇT p2q m pϕq ˇˇď Cph m `τm q. (81)
Besides, mimicking the calculations in the proof of Lemma B.2, one readily shows that v hmτm tends to v in L 2 pQ T q as m goes to `8. Together with (65) and with the uniform convergence of ϕ hmτm towards ϕ stemming from its regularity, this allows to pass to the limit in T p1q m pϕq:

T p1q m pϕq ÝÑ mÑ`8 ij Q T v ¨∇u ϕ dxdt,
concluding the proof of (78) and thus of Proposition 6.4.

A Properties of discrete differential and interpolation operators for the 2D case

We give some properties related to the differential and interpolation operators.

Proposition A.1. We have the following properties:

For any v P p H M pΩq, pδ y ˝δx q v " pδ x ˝δy q v. (82)

For any v P r H M pΩq, pδ x ˝δẙ q v " pδ ẙ ˝δx q v.

Proof. The proof is direct from the above definitions of spaces and operators.

" }δ xy v} 2

L 2 pΩq (5)
" }δ yx v} 2 L 2 pΩq .

Consequently, from the definitions ( 6) and ( 7), (86) holds.

Proposition A.4. Let assume that the mesh M is uniform in each direction in the sense of Definition 4. We then deduce from (85) that

ż Ω pv h ¨∇h u h q ∆ h u h dx " ´żΩ ∇ h pv h ¨∇h u h q ¨∇h u h dx. (99)
We are now in position to state the following technical lemma to be used for the numerical analysis of the scheme (41).

Lemma B.1. For any u h P H M pΩq and v h " pv 1,h , v 2,h q P V E,0 pΩq we have:

ż Ω ∇ h pv h ¨∇h u h q ¨∇h u h dx " T p1,2q 1
`T p2,2q The combination of (107), ( 108) and ( 109) gives (104).

Figure 1 :

 1 Figure 1: Notations for the grid cells of M and x M in the case d " 1.

Figure 2 :

 2 Figure 2: Notations for the grid cells of M, x M, Ă M and Ď M in the case d " 2.

Figure 3 :

 3 Figure 3: Configuration J "sα, βrĂ I with α ą x iα`1 2 and β ă x i β `1 2 .

[ 4 ,y

 4 Section 4.2]. Fix ą 0, then the triangle inequality gives uppx, yq, tq|dydsdt ď A `B m `C m , uppx, yq, tq ´Bx uppx `s, yq, tq|dydsdt, B m " |B x uppx `s, yq, tq ´δx u hmτm ppx `s, yq, tq|dydsdt, C m " |δ x u hmτm ppx `s, yq, tq|dydsdt.

  `Jhm pu hmτm q `|∇ hm u hmτm | 2 ˘|ϕ hmτm ´ϕ| dxdt `Jhm pu hmτm q ´|∇ hm u hmτm | 2 ˘ϕhmτm dxdt Due to the regularity of ϕ and to the boundedness in L 1 pQ T q of J hm pu hmτm q and |∇ hm u hmτm | 2 , we infer that |R m pϕq| ď Cph m `τm q ÝÑ

	and							ij
		S m pϕq "
								Q T
																	mÑ`8	0.	(69)
	Since		ż Ci,j	|∇ hm u n hm | 2 dx "	1 2	σPEi,j ÿ	a σ	`un i,j	´un i,j,σ ˘2 ,
	one deduces from the definition (43) of J hm pu hmτm q that
	|S m pϕq| "	ÿ n"1 ˇˇˇˇˇN T ,m	τ m	ÿ Ci,j PMm	ϕ n i,j	ÿ σPEi,j	a σ	"	i,j,σ ´`u n	´un i,j	˘`¯2	´1 2	`un i,j,σ	i,j ´un	˘2	ˇˇˇˇď
		1 2	N T ,m n"1 ÿ	τ m	Ci,j PMm ÿ	σPEm ÿ	a σ	`un i,j,σ	´un i,j	i,j ˘2 ˇˇϕ n	i,j,σ ´ϕn	ˇď
				N T ,m
		Ch m	ÿ		τ m	ÿ		ÿ	a σ	`un i,j,σ
				n"1			Ci,j PMm	σPEm
	with				ij
	R m pϕq "
					Q T

  Proposition A.2. The following discrete Stokes formula hold: For any pv, wq P H M pΩq ˆp H 0 M pΩq,Proof. The proof is direct from the above definitions of spaces and operators.

			ż
				v δ x w "	´żΩ	δ x v w.
			Ω
			ż
	For any pv, wq P H M pΩq ˆr H 0 M pΩq,		v δ ẙ w "	´żΩ	δ y v w.	(83)
			Ω
			ż
	For any pv, wq P s H x,0 M pΩq ˆr H M pΩq,		δ x v w "	´żΩ	v δ x w.	(84)
			Ω
			ż
	For any pv, wq P s H y,0 M pΩq ˆp H M pΩq,		δ ẙ v w "	´żΩ	v δ y w.
			Ω
			ż
	For any pv, wq P pH M pΩqq 2 ,		v ∆ h w "	´żΩ	∇ h v ¨∇h w. (85)
			Ω
	(83) "	´żΩ	pδ y	˝δx ˝δx qv δ y v dx
	(82) "	´żΩ	pδ x ˝δy ˝δx qv δ y v dx
	" (84)	ż

Consequently, we can also verify the following norms equality: Proposition A.3. For any v P H M pΩq, we have:

}∇ 2 h v} L 2 pΩq " }∆ h v} L 2 pΩq .

(86)

Proof. Let us consider v P H M pΩq. We write:

ż Ω δ xx v δ yy v dx "

ż Ω pδ x ˝δx qv pδ ẙ ˝δy qv dx Ω pδ y ˝δx qv pδ x ˝δy qv dx

  Proof. The proof is direct from the above definitions of spaces and operators.Proposition A.5. Let assume that the mesh M is uniform in each direction in the sense of Definition 4.3. Then: For any pv, wq P H M pΩq ˆp H M pΩq, For any pv, wq P H M pΩq ˆr H M pΩq, For any pv, wq P HM pΩq ˆp H M pΩq, For any pv, wq P HM pΩq ˆr H M pΩq, Proof. The proof is direct from the above definitions of spaces and operators. Proposition A.6. Let assume that the mesh M is uniform in each direction in the sense of Definition 4.3. Then: For any v P p H M pΩq, }π x pvq} L 4 pΩq ď }v} L 4 pΩq , For any v P r H M pΩq, }π ẙ pvq} L 4 pΩq ď }v} L 4 pΩq , For any v P r H M pΩq, }π x pvq} L 4 pΩq ď }v} L 4 pΩq , For any v P p H M pΩq, }π y pvq} L 4 pΩq ď }v} L 4 pΩq .

	3. Then: For any v P p H M pΩq, pδ x For any v P r H M pΩq, pδ x For any v P p H ż ˝πx qv " pπ x ˝πẙ qv " pπ ẙ ˝δx qv, ˝δx qv, Ω π x pvq w dx " ż Ω ż Ω π y pvq w dx " ż Ω ż Ω π ẙ pvq w dx " ż Ω ż Ω ż Ω π x pvq w dx "	(87) (88) v π x pwq dx (92) v π ẙ pwq dx (93) v π y pwq dx (94) v π x pwq dx (95)

M pΩq, pδ x ˝πy qv " pπ y ˝δx qv,

For any v P r H M pΩq, pδ ẙ ˝πx qv " pπ x ˝δẙ qv,

(89)

For any pv, wq P p p

H M pΩqq 2 , δ x pvwq " π x v δ x w `δx v π x w

(90)

For any pv, wq P p r H M pΩqq 2 , δ x pvwq " π x v δ x w `δx v π x w (91)

  pδ x v 1,h ¨πx pδ x u h qq ¨δx u h dx, pδ x v 2,h ¨πx pδ y u h qq ¨δx u h dx, pδ y v 1,h ¨πy pδ x u h qq ¨δy u h dx, pδ x pv 1,h δ x u h qq ¨δx u h dx and T pδ x pv 2,h δ y u h qq ¨δx u h dx. pπ x v 1,h q `δẙ pπ x v 2,h qspδ x u h q 2 dx Lemma B.2. Let v P V 0 pΩq X H 2 pΩq 2 , and let v h P V E,0 pΩq be defined by (38), then}v h } L 2 pΩq 2 ď }v} L 2 pΩq 2 `h2 2 ´}B xx v 1 } 2 L 2 pΩq 2 `}B yy v 2 } 2Moreover, we have the following estimate on ∇ h v h defined by (4):}∇ h v h } L 2 pΩq 2ˆ2 ď }∇v} L 2 pΩq 2ˆ2 `h}B x B y v} L 2 pΩq 2 .and by v 1 , v 2 and v h the corresponding elements in p H M pΩq, r H M pΩq and H M pΩq. Then Jensen's inequality gives that}v h } L 2 pΩq 2 ď }v} L 2 pΩq 2 .So (103) holds true provided that}v h ´vh } L 2 pΩq 2 ď h 2 2 ´}B xx v 1 } 2 L 2 pΩq 2 `}B yy v 2 } 2 ´ξ B xx v 1 px i`1{2 `s, yqdsdξdxdy. |B xx v 1 pxq| dx,hence the Cauchy-Schwarz inequality provides|v i`1{2,j ´vi`1{2,j | 2 ď h 3Therefore, (105) holds true, and thus (103) too.We now focus on inequality (104). Let us first show some control on the first diagonal term of ∇ h v h , the second being similar. Let C i,j P M, thenδ x v 1,h | C i,j " 1 h x h y ż y j`1{2 y j´1{2 `v1 px i`1{2 , yq ´v1 px i´1{2 , yq ˘dy " 1 h x h y B x v 1 px, yqdx. |B x v 1 px, yq| 2 dx,and thus, after summing up over C i,j P M, that}δ x v 1,h } L 2 pΩq ď }B x v 1 } L 2 pΩq , }δ ẙ v 2,h } L 2 pΩq ď }B y v 2 } L 2 pΩq . (106)Let us focus now on the extra-diagonal terms, and particularly on δ y v 1,h , the case of δ x v 2,h being similar. We denote by δ y v 1,h the element of H M pΩq defined byδ y v 1,h | C i`1{2,j`1{2 " 1 h x h y ij C i`1{2,j`1{2 B y v 1 pxqdx, @ C i`1{2,j`1{2 P M,then owing to Jensen's inequality, we get that}δ y v 1,h } L 2 pΩq ď }B y v 1 } L 2 pΩq . `δy v 1,h ´δy v 1,h ˘|C i`1{2,j`1{2 B y pv 1 px i`1{2 , y `sq ´v1 px, y `sqqdxdyds x B y v 1 pξ, y `sqdξdxdyds, whence ˇˇδ y v 1,h ´δy v 1,h ˇˇ| C i`1{2,j`1{2 ď 1 h y |B x B y v 1 pxq| 2 dx, leading to }δ y v 1,h ´δy v 1,h } L 2 pΩq ď h x }B x B y v 1 } L 2 pΩq(108)after summation over i P v1, N ´1w and j P v1, M ´1w. Similarly, we get that }δ x v 2,h } L 2 pΩq ď }B x v 2 } L 2 pΩq and }δ x v 2,h ´δx v 2,h } L 2 pΩq ď h x }B x B y v 2 } L 2 pΩq . (109)

												¯1{2
											L 2 pΩq 2	.	(105)
	To establish (105), let us remark that
	v i`1{2,j ´vi`1{2,j "	1 h x h y	ij	pv 1 px, yq ´v1 px i`1{2 , yqqdxdy
						C i`1{2,j						(107)
	with: On the other hand,	"	ż 1 h x h y	ij C i`1{2,j	ż hx{2 0	ż ξ	1		`T p1,2q 2	`T p2,2q 2	,	(100)
	T 1 p1,2q T p2,2q 1 T p1,2q 2 |v i`1{2,j ´vi`1{2,j | ď " " ż " ż ż hy{2 As a consequence ż yj`1 ż xi`1 1 " h x h 2 y ´hy{2 yj xi ż hy{2 ż yj`1 h x 2h y ż xi`1 ij C i`1{2,j ż x 1 " h x h 2 y ´hy{2 yj xi x i`1{2	to get that
						ż
	and Finally, summing up (101)-(102) and using (87) and (89) yields T p2,2q 2 C i`1{2,j " ż T ij p2,1q 1 " ´1 2 x 4h y |B xx v 1 pxq| 2 dx.	(102)
	T 1 p1,1q Finally, summing up over C i`1{2,j P x `T p2,1q 1 " ż M leads to ´1 2 " ´1 2 ż }v 1,h ´v1,h } 2 L 2 pΩq ď h 4 x }B xx v 1 } 2 L 2 pΩq , | C i`1{2,j`1{2 ij h x ď h y 4 C i`1{2,j`1{2
	p1,2q 1 thanks to similar calculations. Consequently (100) holds. `T p2,2q 1 , as well as T 2 " T 2 p1,2q whereas similar computations yield }v 2,h ´v2,h } 2 L 2 pΩq ď h 4 y 4 }B yy v 2 } 2 L 2 pΩq .	`T p2,2q 2
									p1q 1	`T p2q 1	splits into
												¯1{2
	T 1 p1q	"	ż					1 p2q	"	ż	L 2 pΩq 2	(103)
												ij
	Now from (90) and (91), these expressions further decompose as	(104)
												Ci,j
	T 1 p1q Proof. Let us start by establishing (103). Denote by " T p1,1q 1 `T p1,2q 1 and T p2q 1 " T 1 p2,1q with T p1,2q 1 and T p2,2q 1 as in the statement of Lemma B.1 above, and `T p2,2q 1 T ż v i`1{2,j " 1 h x h y ij C i`1{2,j v 1 dx, v i,j`1{2 " ij Then we deduce from Jensen's inequality that 1 h x h y C i,j`1{2 v 2 dx, ˇˇδ x v 1,h | C i,j 1 ij ˇˇ2 ď h x h y p1,1q 1 " Ci,j
					ż
			T 1 p2,1q	"

Ω π x Ω π ẙ Ω π x Ω π y pδ ẙ v 2,h

¨πẙ pδ y u h qq ¨δy u h dx.

Proof. From the definition of the discrete gradient ∇ h , we have

ż Ω ∇ h pv h ¨∇h u h q ¨∇h u h dx " T 1 `T2 with T 1 " ż Ω δ x pv h ¨∇h u h q ¨δx u h dx and T 2 " ż Ω δ y pv h ¨∇h u h q ¨δy u h dx

Moreover from (96), (

87

) and (88), the term

T 1 " T Ω π x Ω π ẙ Ω π x pπ x pv 1,h q δ xx u h q ¨δx u h dx, Ω π ẙ pπ x pv 2,h q pδ x ˝δy qu h q ¨δx u h dx. Ω δ ẙ pπ x v 2,h qpδ x u h q 2 dx. Ω rδ x Ω rπ x pδ x v 1,h `δẙ v 2,h qspδ x u h q 2 dx " 0 since v h P V E,0 pΩq. Therefore T 1 " T B ij C i`1{2,j`1{2 |B x B y v 1 pxq| dx.

Applying Jensen's inequality, we obtain that ˇˇδ y v 1,h ´δy v 1,h ˇˇ2

" pδ y u h q i,M `1 2 " 0, one can proceed as for T p1,1q 1

Acknowledgments

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).

We then infer from the lower semi-continuity of the norm for the weak convergence (63) that

For the second term B m , the strong convergence of δ x u hmτm towards B x u in L 2 pQ T q (thus also in L 1 pQ T q) stated in (66) implies that lim mÑ`8

For the third term C m , we use the fact that u n 1,j " u n 0,j to write that

`un ´1,j ˇˇ.

Then Cauchy-Schwarz inequality provides that

The first term in the above right-hand side can be overestimated by

Hence we infer from (50) that lim sup

,p2q m , we first use the elementary inequality r a b sb ď a `b to write

Therefore, we obtain that

and thus that lim sup

Combining ( 72)-( 76) in (71) yields

for any ą 0, whence the desired result.

It only remains to prove the following proposition to conclude the proof of Theorem 4.4. Proposition 6.4. Let u be a limit value of pu hmτm q m as in Section 6.1, then u is a weak solution to (30) in the sense of Definition 4.1.

Proof. With Lemma 6.3 in addition to the compactness result stated in Section 6.1, we know that the limit u belongs to the right space to be a weak solution. We only have to check that (30a) is fulfilled by the limit u. To this end, denote by

or i P v1, N m w, j P v1, M m w, and n P v1, N T,m w, then define W hmτm by

if px, tq P C i,j ˆpt n´1 , t n s.

The scheme (41) then rewrites under the compact form

Due to (61), ( 62) and (63), we can easily check that

u∆u weakly in L 2 pQ T q.

Bearing in mind (64) and Lemma 6.2, it only remains to prove the

to pass to the limit in (77) to recover (30a).

Proof. The proof is direct from the above definitions of spaces, operators and the Young inequality.

B Some technical lemmas

Prior to the statement of Lemma B.1 to which this section is devoted, one needs to define some quantities. For the convection term, we define v h ¨∇h u h P H M pΩq by setting v h ¨∇h u h " π x pv 1,h δ x u h q `πẙ pv 2,h δ y u h q.

(96)

A simple calculation gives

with the centered choice

Finally, for the upwind choice used in the scheme (41), we have:

`ÿ σPEi,j v i,j,σ pu i,j,σ,`´ui,j,σ,c q.

(98)

Since pδ x u h q 1 2 ,j " pδ x u h q N `1 2 ,j " 0, we can write:

pπ x pv 1,h qq i,j ppδ x u h q i`1 2 ,j ´pδ x u h q i´1 2 ,j qpδ x u h q i´1 2 ,j `hy 2

pπ x pv 1,h qq i´1,j ppδ x u h q i´1 2 ,j ´pδ x u h q i´3 2 ,j qpδ x u h q i´1 2 ,j

As highlighted in (5), δ x and δ y commute, so that

ż Ω π ẙ pπ x pv 2,h q pδ y ˝δx qu h q ¨δx u h dx.

Since pδ y u h q i, 1