Finite Volume Scheme and Renormalized Solutions for a Noncoercive Elliptic Problem with Measure Data

: The objective of this paper is to show that the approximate solution, by the ﬁnite volumes method, converges to the renormalized solution of elliptic problems with measure data. The methods used are a priori estimates and density arguments. In the ﬁrst part


Introduction
In this work, we consider the discretization by the cellcentered finite volume method of the following convectiondiffusion problem: where Ω is an open bounded polygonal subset of b ∈ L 2 (Ω), b ≥ 0 and µ is a diffuse measure. The space of bounded Radon measures is denote by M q b (Ω), with 1 q + 1 p = 1. J. Carrillo and M. Chipot proved that for µ ∈ M q b (Ω), there exists f ∈ L 1 (Ω) and F ∈ (L p (Ω)) d such that µ = f − divF , see [2].
For the study of problem (1), the obstacles encountered are the noncoercive character of the operator u −→ −∆u + div(vu) + bu and the measure data.
Recall that a renormalized solution of (1) is a measurable function u defined from Ω to R, such that u is finite a.e. in Ω and with T k the truncate function at height k (see Figure 1 below). Since h has a compact support, each term of (4) is well defined. Recently, Botti et al used the finite volume method to solve Poroelasticity problems using high order operators, see [3,4]. The Hybrid High-Order method has also been studied by several authors, see [6,9]. The existence and the uniqueness of a renormalized solution to (1) for L 1 data and b ∈ L 2 (Ω) is proved by M. Ben Cheick and O. Guibé, see [1]. Eymard et al. [11] studied problem (1) with bounded measure data and b ∈ L 2 (Ω). In the present case, it is taken a diffuse measure data instead of L 1 -data, which was considered by S. Leclavier, see [14]. Passing to the limit in a "renormalized discrete version" is the main originality in the present paper, this is to say that a discrete version of ϕh(u) is taken as test function in the finite volume scheme. A first difficulty is to establish a discrete version of the estimate on the energy (3). Moreover it is worth noting that in (4) all the terms are "truncated" while a discrete version of ϕh(u) in the finite volume scheme leads to some residual terms which are not "truncated". The second difficulty is to deal with the diffuse measure data.
The paper is organized as follows. In Section 2, we present the finite volume scheme and the properties of the discrete gradient. Section 3 is devoted to prove several estimates, especially the discrete equivalent to (4) which is crucial to pass to the limit in the finite volume scheme. In Section 4, the proof of the convergence of the cell-centered finite volume scheme via a density argument is concerned. A brief conclusion closes this work in the last section.

Finite Volume Scheme
Let us define the admissibility mesh in the present work, see [14].
A family T of related subsets of Ω ⊂ R d is called a mesh. Any K ∈ T is called control volume. We impose that every K ∈ T is opened, the union of the T is Ω and the interface is in some hyperplane. For K, L ∈ T two distinct control volumes, their interface is denoted by K/L := K ∩ L . Let K ∈ T , we can write (N for "neighbour"): N (K) = {L ∈ T ; L / ∈ K, K/L = ∅}, the set of neighbour of K and ∂K = L∈N (K) K/L, the edge of K. Finally, the Lebesgue measure in d-dimensional is denoted by |K| and by |∂K| (respectively K/L) for the (d − 1)-dimensional measure of ∂K (resp. of K/L).
Thus, set E a finite family of disjoint subsets of Ω contained in affine hyperplanes, called the "edges", and set P = (x K ) K∈T a family of points in Ω such that : 1. each σ ∈ E is a non-empty open subset of ∂K, for some K ∈ T , 2. by denoting E(K) = {σ ∈ E; σ ∈ ∂K}, one has ∂K = σ∈E(K) σ for all K ∈ T , 3. for all K = L in T , either the measure of K ∩ L is null or K ∩ L = σ for all σ ∈ E, that we denote then σ = K/L, 4. for all K ∈ T , x k is in the interior of K, 5. for all σ = K/L ∈ E, the line [x K , x L ] intersects and is orthogonal to σ, 6. for all σ ∈ E, σ ⊂ ∂Ω∩∂K, the line which is orthogonal to σ and going through x k intercepts σ. We denote by |K| (resp. |σ|) the Lebesgue measure of K ∈ T (resp. of σ ∈ E ). The unit normal to σ ∈ E(K) outward to K is denoted by η K,σ . E int (resp. E ext ) is defined as the set of interior (resp. the boundary) edges. For all K ∈ T and σ ∈ E(K) , we denote d K,σ the Euclidean distance between x k and σ.
For any σ ∈ E, d σ is defined by d σ = d K,σ + d L,σ , if σ = K/L ∈ E int (in which case d σ is the Euclidean distance between x K and x L ) and The size of the mesh, denoted by h T , is defined by h T = sup There exists ζ > 0 such that for all K ∈ T and for all σ ∈ E K , In the sequel, the discrete W 1,q 0 norm and the discrete versions of Poincaré and Sobolev inequalities will be useful to solve the problem (see [7]).
where v K = χ K v. Definition 2.2. (Discrete finite volume gradient) For all K ∈ T and for all σ ∈ E(K), we define the volume D K,σ as the cone of basis σ and of opposite vertex x K . Then, we define the "diamond-cell" D σ by: For any v T ∈ X(T m ) and notice that |D σ | = |σ| d σ d , the discrete gradient ∇ T v T is defined by: T be an admissible mesh and ζ > 0 satisfying for all K ∈ T and all σ ∈ E(K) , d K,σ ≥ ζd σ , Then, with q * = dq Before writing the finite volume scheme, let us define a discrete finite volume gradient (see [13])).
Lemma 2.1. (Weak convergence of the finite volume gradient) Let (T m ) m≥1 be a sequence of admissible meshes such that there exists ζ > 0 satisfying for all m > 1, for all K ∈ T and for all σ ∈ E(K), d K,σ ≥ ζd σ , and such that h Tm −→ 0. Let v Tm ∈ X(T m ) and let us assume that there exists α ∈ [1, +∞[ and C > 0 such that v Tm 1,α,Tm ≤ C, and v Tm and that L 1 (Ω) converges in v ∈ W 1,α 0 (Ω). Then ∇T m v Tm converges to ∇v weakly to L α (Ω) d . Let T be an admissible mesh, we can define the finite volume discretization of (1). For K ∈ T and σ ∈ E(K), So, we can write the scheme (1) as following: For all K ∈ T , with We denote by u σ,− the downstream choice of u which is such that

Estimations
In this section, we first establish in Proposition 3.1 an estimate on ln(1+|u T |) which is crucial to control the measure of the set {|u T | > n}. Then, we show in Proposition 3.2 an estimate on T n (u T ) and the convergence of T n (u T ) to T n (u). Finally, we prove in Proposition 3.3 a discrete version of the decay of the energy.
Proposition 3.1. (see [14]) Let T be an admissible mesh. If u T = (u K ) K∈T is a solution to (10), then where C(µ, Ω) is a constant depending on µ and Ω. Let us state an easy corollary, which is used in the proof of the estimate of Proposition 3.2.
Corollary 3.1. Let T be an admissible mesh. if u T = (u K ) K∈T is a solution to (10) and, for n > 0, E n = {|u T | > n} , then there exists C > 0 only depending on (Ω, v, f, d, p) such that Proposition 3.2. (Estimation on T n (u T ) ) Let T be an admissible mesh. if u T = (u K ) K∈T is a solution to (10), then there exists C > 0 only depending on (Ω, v, µ, n, d) such that Moreover, if (T m ) m≥1 is a sequence of admissible meshes such that there exists ζ > 0 satisfying for all m ≤ 1, for all K ∈ T and for all σ ∈ E(K) , d Kσ ζd σ , there exists a measurable function u finite a.e. in Ω such that, up to a sub-sequence T n (u Tm ) converges to T n (u) weakly in H 1 0 (Ω), strongly in L 2 (Ω) and a.e. in Ω.
Proof The proof is divided into two steps. In Step 1 we derive the estimate (15) on the truncate on u T . The step 2 is devoted to extract a Cauchy sub-sequences in measure.
Step 1: Estimation on T n (u T ) Multiplying each equation of the scheme (2.6) by T n (u K ), summing over each control volume and reordering the sum, we Since b is nonegative and since rT n (r) 0 ∀r , we notice that S 3 is nonnegative. Moreover, since T n is bounded by n, we deduce that For the term S 5 , Hölder inequality and relation (5) yield Therefore, S 2 can be rewritten as, The subset A of edges is defined by (see [10]), and since T n is non decreasing we have −S 2 = σ∈E |σ||v K,σ |u σ,+ (T n (u σ,− ) − T n (u σ,+ )).

It follows that
and we can deduce that Therefore, using again the fact that T n is 1-Lipschitz, we can write : Applying Lemma 2.1 and the diagonal process, up to a subsequence still denoted by T m , for any n ≥ 1, there exist v n in H 1 0 (Ω) such that T n (u T ) −→ v n and T n (u T ) v n in the finite volume gradient sense.
Step 2: Up to a subsequence, u T is a Cauchy sequence in measure In this step, we follow the ideas of Dal Maso et al. to show that u Tm converges a.e. to u (see [8]). Let ω > 0. For all n > 0 and all sequences (T m ) m≥1 and (T p ) p≥1 of admissible meshes, we have Let ε > 0 fixed. By (14), let n > 0 such that, for all admissible meshes T m and T p , Once n is chosen, we deduce from Step 1 that T n (u Tm ) is a Cauchy sequence in measure, thus Therefore, we deduce that ∀h Tm , h Tp < h 0 , meas({|u Tm − u Tp | > ω}) < ε.
Hence (u Tm ) is a Cauchy sequence in measure. Consequently, up to a subsequence still indexed by T m , there exists a measurable function u such that u Tm −→ u a.e. in Ω. Due to Corollary 3.1, u is finite a.e. in Ω. Moreover from convergences obtained in Step 1 we get that In the following proposition we prove a uniform estimate on the truncated energy of u T (see (19)) which is crucial to pass to the limit in the approximate problem. We explicitly observe that (19) is the discrete version of (3) which is imposed in the definition of the renormalized solution for elliptic equation with measure data. As in the continuous case (19) is related to the regularity of f : f ∈ L 1 (Ω) and does not charge any zero-Lebesgue set. If we replace div(vu), we also have to uniformly control the discrete version of 1 n Ω vu∇T n (u)dx which is stated in (20). Proposition 3.3. (Discrete estimate on the energy) Let (T m )m ≥ 1 be a sequence of admissible meshes such that there exists ζ > 0 satisfying ∀m 1, ∀K ∈ T and ∀σ ∈ E(K), d K,σ ≥ ζd σ .
If u Tm = (u K ) K∈Tm is a solution to (10), then where u L = 0 if σ ∈ E ext , and Proof We first establish (19). Let T be an admissible mesh and let u T be a solution of (10). Multiplying each equation of the scheme by T n (u K ) n , summing on K ∈ T and gathering by edges lead to T 1 + T 2 + T 3 = T 4 − T 5 with Since b is non-negative and since rT n (r) ≥ 0 ∀r , we get T 3 ≥ 0 . Due to the definition of u T we have In view of the point-wise convergence of u T to u, we obtain that T n (u T ) converges to T n (u) a.e. and weak Since u is finite a.e. in Ω, T n (u) n converges to 0 a.e. and in L ∞ weak, and since f belongs to L 1 (Ω), the Lebesgue dominated convergence theorem implies that For the term T 5 , using Hölder inequality, relation (5) and Definition 2.2 yield Using the strong convergence in L 2 (Ω) and a.e. in Ω of T n (u T ) to T n (u), and relation (22), give (23) Using the same techniques as S. Leclavier [14], it follows that From (22), we deduce (19). By the same manage used by S. Leclavier [14], we prove (20).
The following corollary is useful to pass to the limit in the diffusion term.
If u Tm = (u K ) K∈Tm is a solution to (10), then

Convergence Analysis
Let us now state the main result of this paper. Theorem 4.1. If T is an admissible mesh, then there exists a unique solution to (10). If (T m ) m≥1 is a sequence of admissible meshes such that there exists ξ > 0 satisfying for all m ≥ 1 for all K ∈ T and all σ ∈ E(K), d K,σ ≥ ξd σ , and such that h Tm → 0 , then if u Tm = (u K ) K∈Tm is the solution to (10) with T = T m , u Tm converges to u in the sense that for all n > 0, T n (u Tm ) converges weakly to T n (u) in H 1 0 (Ω), when u is the unique renormalized solution of (1).
Before proving Theorem 4.1, we recall the following convergence result concerning the function (h n ) defined, for any n ≥ 1, by (see [14]) Lemma 4.1. Let (T m ) m≥1 be a sequence of admissible meshes such that there exists ξ > 0 satisfying for all m ≥ 1, for all K ∈ T and for all σ ∈ E(K), d K,σ ≥ ξd σ . Let u Tm ∈ X(T m ) be a sequence of solution of (10). We define the function h n by σ ∈ E ,∀x ∈ D σ , h n (x) = hn(x K )+hn(x L ) 2 , then h n → h n (u) in L q (Ω), ∀q ∈ [2, +∞[ where h Tm → 0, where u is the limit of u Tm .
Proof Proof Proof of Theorem 4.1 The proof consists into two steps. In Step 1 we prove the existence and the uniqueness of the solution of (10). Concerning the uniqueness of the renormalized solution, the proof is done by Ouédraogo et al. [16]. To prove the second point, we adapts one uniqueness techniques developed in the continuous case by several authors (see [5,12]). It is worth noting that we use here a different method to the one developed by J. Droniou et al. [10]. Using the results of Section 3, Step 2 is devoted to pass to the limit in the scheme. It is worth noting that we take in the scheme a discrete version of what is a test function in the renormalized formulation.

Existence and Uniqueness of the Solution of the Scheme
Since the relation (10) is a linear system of n equations with n unknowns, it is sufficient to show that the solution of the relation (10) with µ = 0 (see [14]).

Convergence
Let ϕ ∈ C ∞ c (Ω) and h n the function defined by (26). We denote by ϕ T the function defined by ϕ K = ϕ K (x K ) for all K ∈ T . Multiplying each equation of the scheme (10) by ϕ(x K )h n (u K ) (which is a discrete version of the test function used in the renormalized formulation), summing over the control volumes and gathering by edges, we get As far as the term V 4 is concerned, by the regularity of ϕ, we have ϕ T → ϕ uniformly on Ω when h T → 0. We now pass to the limit as h T → 0. Since h n (u T ) → h n (u) a.e and L ∞ weak * , ϕ T → ϕ uniformly, |f ϕ T h n (u T )| ≤ C ϕ |f | ∈ L 1 (Ω), the Lebesgue dominated convergence theorem ensures that Due to the definition of ∇ T (·) we get for the term V 5 : To pass to the limit as h T → 0 in (28), the following lemma is useful.
Lemma 4.2. Let T be an admissible mesh, ϕ ∈ C ∞ c (Ω) and h n the function defined by (26).. if u T = (u K ) K∈T is a solution to (10), then there exists C > 0 only depending on (Ω, v, µ, n, d) such that According to Lemma 4.2, ∇ T ϕ T (x T ) h n (u T ) converges to ∇(ϕ(x) h n (u)) weakly in H 1 0 (Ω), as h T → 0. Therefore, passing to the limit in (28) gives In view of the definition of b T , and since b belongs to L 1 (Ω), b T = (b K ) K∈T converges to b in L 1 (Ω) as h T → 0. With already used arguments we can assert that We now study the convergence of the diffusion term. We write Using the same techniques as S. Leclavier [14], it follows that and lim h T →0 For the convection term we have |σ|v K,σ u σ,+ ϕ(x K )(h n (u σ,+ ) − h n (u σ,− )) |σ|v K,σ u σ,+ ϕ(x L )(h n (u σ,+ ) − h n (u σ,− )), Using the same techniques as S. Leclavier [14], it follows that and lim We are now in position to pass to the limit as h T → 0 in the scheme (10). Gathering equations (27) to (35), we can assert that where lim Let h ∈ C 1 c (R) and ψ ∈ C 1 c (Ω) ∩ H 1 0 (Ω). In view of the regularity of T n (u)(see (2)) the function h(u)ψ belongs to F · ∇(ψ h(u) h n (u)) dx ≤ ϕ L ∞ (Ω) ω(n).
Passing to the limit as n → +∞ in the previous inequality yields that : which is Equality (4) in the definition of a renormalized solution. It remains to prove that u satisfies the decay (3) of the truncate energy.
Thanks to the discrete estimate on the energy (19) we get, Since ∇T 2n (u T ) converges weakly in L 2 (Ω) d , we have also which leads to lim n→0 1 n Ω |∇T 2n (u)| 2 dx = 0.
Since the renormalized solution u is unique, we conclude that the whole sequence u Tm converges to u in the sense that for all n > 0, T n (u Tm ) converges weakly to T n (u) in H 1 0 (Ω).

Conclusion
In this paper, the finite volumes method has been used to prove that the approximate solution converges to the renormalized solution of elliptic problems with measure data. A first difficulty is to establish a discrete version of the estimate on the energy (3). Moreover it is worth noting that in (4) all the terms are "truncated" while a discrete version of ϕh(u) in the finite volume scheme leads to some residual terms which are not "truncated". The second difficulty is to deal with the diffuse measure data. Firstly, we presented the finite volume scheme and the properties of the discrete gradient. Secondly, we are proven several estimates, especially the discrete equivalent to (4) which is crucial to pass to the limit in the finite volume scheme. At last, we established the convergence of the cell-centered finite volume scheme via a density argument.