The G method for heterogeneous anisotropic diffusion on general meshes

DOI:10.1051/m2an/2010021 www.esaim-m2an.org

THE G METHOD FOR HETEROGENEOUS ANISOTROPIC DIFFUSION ON GENERAL MESHES

L´ eo Ag´ elas

, Daniele A. Di Pietro

and J´ erˆ ome Droniou

Abstract. In the present work we introduce a new family of cell-centered Finite Volume schemes for anisotropic and heterogeneous diffusion operators inspired by the MPFA L method. A very general framework for the convergence study of finite volume methods is provided and then used to establish the convergence of the new method. Fairly general meshes are covered and a computable sufficient criterion for coercivity is provided. In order to guarantee consistency in the presence of heterogeneous diffusivity, we introduce a non-standard test space inH₀¹(Ω) and prove its density. Thorough assessment on a set of anisotropic heterogeneous problems as well as a comparison with classical multi-point Finite Volume methods is provided.

Mathematics Subject Classification. 65N08, 65N12.

Received November 27, 2008.

Published online March 17, 2010.

1. Introduction

One of the key ingredients for the numerical solution of Darcy equations is the discretization of anisotropic heterogeneous elliptic terms [9]. Significant mathematical effort has therefore been devoted to finding consistent and robust Finite Volume (FV) discretizations of anisotropic heterogeneous elliptic terms on general meshes.

Ideally, a method should (i) be consistent and coercive on general polyhedral meshes as well as robust with respect to the anisotropy and heterogeneity of the permeability tensor; (ii) yield well-conditioned linear systems for which optimal preconditioning strategies can be devised; (iii) have a narrow stencil, both to improve matrix sparseness and to reduce the communication in parallel implementations. The latter requirement speaks in favour of cell-centered methods. However, at present time, no unconditionally coercive and consistent compact stencil cell-centered method has been found. Indeed, although several symmetric methods display unconditional coercivity, they either entail severe mesh restrictions, as in [4], or exhibit very large stencils, as in [6,24].

The so-called Multi Point Flux Approximation (MPFA) methods have been independently introduced in the middle of the 90s by Aavatsmarket al.[2] and Edwards and Rogers [21]. The key idea is to obtain consistency on general meshes at the expense of a larger stencil while preserving the second order convergence of the classical two-points method. As mentioned, however, coercivity only holds under suitable conditions on both the mesh and the permeability tensor. The compact stencil MPFA L method has been proposed by Aavatsmark et al.

Keywords and phrases. Finite volume methods, heterogeneous anisotropic diffusion, MPFA, convergence analysis.

1 IFP, 1 & 4 av. du Bois-Pr´eau, 92852 Rueil-Malmaison Cedex, France. [email protected];

[email protected]

2 Universit´e Montpellier 2, Institut de Math´ematiques et Mod´elisation de Montpellier, CC 051, Place Eug`ene Bataillon, 34095 Montpellier Cedex 05, France. [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2010

[3,5] as an improvement of the MPFA O method of [1] both in terms of stencil size and monotonicity properties.

The convergence of the MPFA O method has been theoretically proved in [4] on two-dimensional quadrilateral grids and in [7,8] on general two- and three-dimensional polyhedral meshes. In [25], the equivalence of multi- points methods with the lowest-order Mixed Finite Element method on matching triangular grids has been pointed out, and local coercivity conditions have been proposed. Other relatively inexpensive methods that deserve being mentioned are those developed in the Mimetic Finite Difference framework of [14–16], as well as the Hybrid Finite Volume scheme of [24] or the Mixed Finite Volume scheme of [19]. The analogies among the three classes of methods have been recently pointed out in [20]. Finally, a unified framework covering both FV and discontinuous Galerkin methods expressed in weak form has recently been introduced in [10] relying on the discrete functional analysis results of [17,23].

In this work we propose a family of cell-centered schemes generalizing the MPFA L method. The key idea is to write the flux through a face as the weighted average of several L-type fluxes. A proper choice of the weights enhances the coercivity of the method, thereby improving its robustness with respect to the skewness of the mesh and to the anisotropy and heterogeneity of the permeability tensor. The provided convergence proof covers more general FV schemes expressed in terms of numerical fluxes and it is inspired by the techniques of Eymard, Gallou¨et and Herbin (see, e.g., [22,24]). The relevant requirements are weak flux consistency and coercivity. Convergence is then obtained using the discrete Sobolev embeddings and Rellich theorem proved in [24], Section 5. Unlike in [10], where methods in weak formulation are considered, we focus here on lower order methods in flux formulation. The interest of flux formulation is that (i) it provides a natural means to implement new methods in traditional two-point FV codes; (ii) it is more natural for a number of multi-points methods and (iii) it allows to further reduce the set of requirements for convergence (flux continuity,e.g., is not needed).

From a practical viewpoint, the proposed method is a good compromise between accuracy, robustness and computational cost. Indeed, the methods of [14–16,20,24] require the introduction of additional face unknowns whose local elimination in terms of cell unknowns is, in general, not possible. While the resulting stability properties are highly appreciable, the increase in computational cost is often not affordable in large industrial simulations. Unconditionally coercive cell-centered methods like the ones of [24], Section 2.2, or of [6] have stencils extending to neighbours of neighbours, resulting in denser matrices and increased memory require- ments. Also, in parallel implementations, two layers of ghost cells are needed to ensure communications among subdomains, resulting in heavily penalized scalability (message passing is still considered a bottle-neck when it comes to large industrial cases). More compact methods like the MPFA L method have up to now been based on (sophisticated) heuristics rather than on a extensive mathematical analysis. To the best of our knowledge, the present work contains the first rigorous convergence proof for the MPFA L method for general meshes and arbitrary heterogeneous anisotropic diffusion tensors. The aim of this paper is also to identify a minimal set of requirements for convergence and investigate the benefits of a deeper mathematical comprehension. The resulting MPFA G method outperforms the original version of [3,5] on a number of representative test cases modelling some of the difficulties encountered in industrial simulations.

In order to avoid artificial regularity assumptions on the permeability tensor in the consistency proof, we have introduced a permeability dependent test spaceQcomposed of continuous functions with possibly discontinuous gradients but continuous fluxes. This space is proved to be dense in H₀¹(Ω) following the ideas of [18]. To the best of our knowledge, the idea of selecting a problem-taylored test space as well as the density proof are new;

also, the density results forQis a general tool of independent interest.

The work is organized as follows: in Section 2we present a general convergence study based on a minimal set of requirements; the analysis applies to fairly general FV methods expressed in terms of numerical fluxes. In Section3we present the G method, a generalization of the MPFA L scheme, and show that it fits in the analysis framework of Section2; Section4 is devoted to numerical tests. The performances of the proposed method are evaluated on a set of anisotropic and heterogeneous benchmarks on general meshes. A comparison against the MPFA O and L methods as well as against a variant of the symmetric unconditionally coercive scheme of [24], Section 2.2 (hereafter labeled Success), is provided.

2. Abstract framework 2.1. Model problem

Let Ω ⊂ R^d, d ≥ 1, be an open bounded connected polygonal domain with boundary ∂Ω and let P_Ω ^def= {Ωi}i=1...NΩ denote a finite partition of Ω into open connected disjoint polygonal subsets. Let Λ be a symmetric tensor-valued function such that (s.t.) (i) Λ_|Ωi ∈[C²(Ω_i)]^d,d for alli= 1. . . N_Ω and (ii) there exist 0< α₀ <

β₀<+∞s.t., for almost every (a.e.)x∈Ω, the spectrum of Λ(x) is contained in [α₀, β₀]. Consider the following

problem:

∇·(−Λ∇u) =f in Ω,

u= 0 on∂Ω, (2.1)

where f ∈L^r(Ω) with r >1 ifd= 2 andr= _d+2^2d ifd > 2. The existence and uniqueness of a weak solution u∈H₀¹(Ω) of problem (2.1) is a classical result.

Remark 2.1. Other standard types of boundary conditions can be considered. However, for easiness of pre- sentation, we have preferred to consider the simpler homogeneous Dirichlet case.

In what follows, we shall provide the definition of a FV discretization of problem (2.1) as well as an analysis framework covering fairly general (possibly nonconforming) polygonal meshes.

Definition 2.1 (admissible family of discretizations). An admissible family of finite volume discretizations {Dn}n∈N is a tripletDn = (Tn,En,Pn), where

(i) Tn is a finite family of non-empty connected open disjoint subsets of Ω (thecells or control volumes) s.t.

Ω =∪K∈TnK and Tn is compatible with P_Ω (each cell is contained in one element of the partition P_Ω).

For allK∈ Tn, we denote by m_K >0 its d-dimensional measure (thevolume) and let∂K^def= K\K;

(ii) En is a finite family of subsets of Ω (the faces) s.t., for all σ∈ En, σ is a non-empty closed subset of a hyperplane of R^d with (d−1)-dimensional measure m_σ>0 (the area), and s.t. the intersection of two different faces has zero (d−1)-dimensional measure. We assume that, for allK∈ T_n, there exists a subset EK ofEn such that ∂K=∪σ∈EKσ. For a given σ∈ En, either Tσdef

= {K∈ Tn|σ∈ EK} has exactly one element and then σ⊂∂Ω (boundary face) or Tσ has exactly two elements (inner face); the sets of inner and boundary faces are denoted byEn,int andEn,ext respectively;

(iii) Pn = {xK}K∈Tn is a family of points of Ω indexed by Tn (the cell centers) s.t. x_K ∈ K and K is star-shaped with respect to x_K. For all K ∈ Tn and for all σ ∈ EK we denote by d_K,σ the Euclidean distance between x_K and the hyperplane supporting σ, and suppose that there exist 0< ₁<+∞ and 0< ₂<+∞independent ofns.t.

K∈Tminn, σ∈EK

d_K,σ

diam(K)≥₁, min

σ∈En,int,Tσ={K,L}

min(d_K,σ, d_L,σ)

max(d_K,σ, d_L,σ) ≥₂. (2.2) Figure1(a)presents an example of admissible mesh in two space dimensions. Definition2.1also establishes refinement procedure to obtain admissible mesh families from an initial coarse grid. By items (ii) and (iii), and since ^mσd_d^K,σ is the measure of the convex hullK,σ ofx_K andσ(see Fig.1(b)), the following geometric relation holds:

∀K∈ Tn,

σ∈EK

m_σd_K,σ=dm_K. (2.3)

The size of the discretization is defined byh_Dn ^def= sup_K∈Tndiam(K). For allK∈ Tn andσ∈ EK, we denote byn_K,σthe unit vector normal toσoutward toK. For allK∈ Tn, we set Λ_K ^def= _m¹

K

KΛ(x) dx. The Euclidean norm of a vectorx∈Rⁿ will be denoted by|x|^def= √

x·x, while, for a matriceA∈R^n,n,|A|will denote the norm

(a) Admissible mesh

K σ

K,σ

x_K

(b) Pyramid convex hull ofxK andσ

Figure 1. Example of admissible mesh and pyramid convex hull of x_K andσford= 2.

induced by the scalar product ofRⁿ ,i.e., |A|^def= sup_x∈Rd

|Ax|

|x| . The vector space of bounded linear operators fromE to F will be denoted byL(E;F).

In what follows, when referring to a generic elementDn of an admissible family of discretizations{Dn}n∈N, the subscriptnwill be omitted to alleviate the notation whenever no ambiguity arises. The space of piecewise constant functions onT is defined as

H_T(Ω)^def= {v∈L²(Ω)|v_|K ∈P⁰(K), ∀K∈ T }.

For all v ∈ H_T and for all K ∈ T, v_K will denote the (constant) value of v on K, i.e., v_|K(x) = v_K for all x∈K. In order to endowH_T with a discreteH¹ norm, let, for allσ∈ E,I_σ∈ L(H_T(Ω);P⁰(σ)) denote a trace reconstruction operator s.t., for allv∈H_T(Ω), I_σv= 0 ifσ∈ Eext. We are now ready to define

||v||T,I def

=

K∈T

σ∈EK

m_σ

d_K,σ|Iσv−v_K|² _1/2

.

Remark 2.2. Letγ_σ∈ L(HT(Ω);P⁰(σ)) be s.t.

∀v∈H_T(Ω),

⎧⎨

⎩

γ_σv−v_K

d_K,σ +γ_σv−v_L

d_L,σ = 0 ifσ∈ Eint withTσ={K, L}, γ_σv= 0 ifσ∈ Eext.

Then, for allI_σ ∈ L(HT(Ω);P⁰(σ)) s.t., for allv∈H_T(Ω),I_σv= 0 ifσ∈ Eext,

∀v∈H_T(Ω), v_T,γ ≤ v_T,I. (2.4)

Set, for σ ∈ Eint with Tσ={K, L}, g_σ(y) ^def= _d^mσ

K,σ|y−v_K|²+ _d^mσ

L,σ|y −v_L|². Inequality (2.4) is obtained by noticing thatγ_σvminimizesg_σ and that v²_T,γ=

σ∈Eintg_σ(γ_σv) +

σ∈Eext,Tσ={K} mσ

dK,σ|vK|².

In view of Remark 2.2 and of the special nature of γ_σ, the abridged notation ·T will be used for ·T,γ

whenever possible. For allK∈ T and for allσ∈ EK, letF_K,σ∈ L(H_T(Ω);P⁰(σ)) be a numerical flux function meant to approximate the diffusive flux flowing outKthroughσ. For all (u, v)∈[H_T(Ω)]², define the bilinear form

a_T(u, v)^def=

K∈T

σ∈EK

F_K,σ(u)(I_σv−v_K).

In what follows, we shall consider discretizations of (2.1) of the form Findu∈H_T(Ω) s.t.a_T(u, v) =

Ω

f v for allv∈H_T(Ω). (2.5)

2.2. Convergence analysis

We introduce the discrete gradient reconstruction ∇_D ∈ L(H_T(Ω); [H_T(Ω)]^d) s.t., for all K ∈ T and all v∈H_T(Ω),

∇_Dv_|K = 1 m_K

σ∈EK

m_σ(I_σv−v_K)n_K,σ. (2.6)

For all v ∈H_T and for all K ∈ T, (∇_Dv)_K will denote the (constant) value of ∇_Dv onK, i.e., ∇_Dv_|K(x) = (∇Dv)_K for allx∈K. Equation (2.3) together with Cauchy-Schwarz inequality yield

∇_Dv_[L²_(Ω)]^d ≤√

dv_T,I ∀v∈H_T(Ω). (2.7)

The next result can proved following [24], Section 5.1.2 (indeed, the discrete norms considered are equivalent under the mesh regularity assumptions of Def.2.1):

Lemma 2.1 (discrete Sobolev embeddings). Let D be an element of a family of discretizations matching Definition 2.1. Letq∈[1,+∞)if d= 2, andq∈[1,2d/(d−2)] ifd >2. Then, there existsC₁>0 depending on Ω,q,₁ and₂ but not on ns.t.

u_L^q_(Ω)≤C₁u_T ∀u∈H_T(Ω).

Owing to Remark 2.2, the following theorem can easily be deduced from (2.7) using techniques of [24], Lemmata 5.6–5.7:

Lemma 2.2 (discrete Rellich theorem). Let {Dn}_n∈N be a sequence of admissible discretizations matching Definition 2.1 and s.t. h_Dn → 0 as n→ ∞, and let {vn}_n∈N be a sequence of H_Tn(Ω) s.t. there exists C >0 withvn_Tn,I≤C for alln∈N. Then, there exist a subsequence of {vn}_n∈N and a functionv∈H₀¹(Ω)s.t., as n→ ∞,(i)v_n→v inL^q(Ω) for allq∈[1,2d/(d−2))(and weakly inL^2d/(d−2)(Ω)if d >2);(ii){∇Dnv_n}n∈N

weakly converges to ∇v in [L²(Ω)]^d.

The assumptions yielding convergence of the finite volume scheme are gathered in the following

Assumption 2.1. Let {Dn}n∈N be a family of discretizations matching Definition 2.1 and s.t. h_Dn→0 as n→ ∞. We suppose that

(P1) D is a dense subspace of H₀¹(Ω) s.t. D⊂C₀(Ω)∩C²(P_Ω), where C²(P_Ω) is the set of functions whose restriction toΩ_i,i= 1. . . N_Ω, isC²(Ω_i)andC₀(Ω)denotes the space of continuous functions which vanish on∂Ω. For all ϕ∈D, we denote byϕ_Tn the element of H_Tn(Ω) s.t., for all K∈ Tn,ϕ_Tn|K =ϕ(x_K);

(P2) a_Tn is uniformly coercive, i.e., there isγ₁>0 independent of ns.t.

∀v∈H_Tn(Ω), a_Tn(v, v)≥γ₁v²_Tn,I; (P3) the numerical fluxes are weakly consistent onD,i.e., for allϕ∈D,

Dn(ϕ)^def=

K∈Tn

σ∈EK

d_K,σ

m_σ |F_K,σ(ϕ_Tn)−m_σΛ∇ϕK·n_K,σ|^{2 1}₂

→0 asn→ ∞, (2.8)

where, for all K ∈ T and for all Φ ∈ L¹(K), we have set ΦK def

=

KΦ(x) dx/m_K. For vectorial functions, the notation has to be intended component-wise.

Remark 2.3. Owing to (2.3), Property (P3) holds for strongly consistent numerical fluxes, for which there is C₂ independent ofns.t., for allϕ∈D,

∀K∈ Tn, ∀σ∈ EK, |FK,σ(ϕ_Tn)−m_σΛ∇ϕK·n_K,σ| ≤C₂m_σh_Dn. (2.9) Indeed, owing to (2.3), (2.9) implies _Dn(ϕ)≤C₂√

dm_Ωh_Dn, whence (P3).

Remark 2.4. Finite Volume methods are usually conservative,i.e., for allv ∈H_Tn(Ω) and allσ∈ En,int with Tσ ={K, L},F_K,σ(v) +F_L,σ(v) = 0. This property is not required to prove Theorem2.1below. However, it is usually needed in the proof of (P2) or even in the definition of the numerical fluxes. When conservativity holds, problem (2.5) is equivalent to the (more classical) FV formulation:

Find u∈H_T(Ω) s.t. −

σ∈EK

F_K,σ(u) =

K

f(x) dxfor allK∈ T, and the bilinear forma_T does not depend on the choice of the trace operators{Iσ}σ∈E.

Proposition 2.1 (asymptotic stability of the interpolator). Under Assumption2.1, for allϕ∈D, ϕ_TT,I≤ 1

γ₁

D(ϕ) +β₀√

d|ϕ|_H¹_(Ω) .

Proof. Letϕ∈D. Owing to (2.6), for allv∈H_T(Ω), the following integration by parts formula holds:

K∈Tn

σ∈EK

m_σΛ∇ϕK·n_K,σ(I_σv−v_K) =

K∈Tn

K

Λ(x)∇ϕ(x)·

1 m_K

σ∈EK

m_σn_K,σ(I_σv−v_K)

dx

=

K∈Tn

K

Λ(x)∇ϕ(x)·(∇Dv)_Kdx=

Ω

Λ(x)∇ϕ(x)·∇Dv(x) dx.

(2.10) The above result together with (P2), Cauchy-Schwarz inequality and (2.7) yield

γ₁ϕ_T²_T,I≤a_T(ϕ_T, ϕ_T) =

K∈T

σ∈EK

d_K,σ

m_σ [F_K,σ(ϕ_T)−m_σΛ∇ϕK·n_K,σ] m_σ

d_K,σ(I_σϕ_T −ϕ_K) +

Ω

Λ(x)∇ϕ(x)·∇Dϕ_T(x) dx

≤ _D(ϕ)ϕ_TT,I+β₀|ϕ|_H¹_(Ω)∇ϕ _T[L2(Ω)]^d≤

D(ϕ) +β₀√

d|ϕ|_H¹_(Ω)

ϕ_TT,I.

Lemma 2.3 (well-posedness). Assume that Assumption2.1holds. Then, problem (2.5)is well-posed for each n∈N, and the solutions u_n∈H_Dn(Ω) satisfy the stability estimate

u_nTn,I ≤ C₁

γ₁fL^r(Ω). (2.11)

Proof. The solvability of problem (2.5) stems from (P2), which guarantees the inversibility of the corresponding linear system for eachn. Using (P2), H¨older’s inequality, Lemma 2.1 and Remark2.2, it is inferred from the integrability assumptions onf that

γ₁un²_Tn,I ≤a_Tn(u_n, u_n) =

Ω

f udx≤ f_L^r_(Ω)un_Lr(Ω)≤C₁f_L^r_(Ω)un_Tn,I,

where we have setr ^def= _r−1^r .

Theorem 2.1 (convergence). Let {Dn}_n∈N be a family of discretizations matching Assumption 2.1 and s.t.

h_Dn → 0 as n → ∞. Then, as n → ∞, the sequence of discrete solutions of problem (2.5), say {un}n∈N, converges to the solutionuof (2.1)inL^q(Ω) for allq∈[1,2d/(d−2)) (and weakly inL^2d/(d−2)(Ω) if d >2).

Proof. Owing to the stability estimate (2.11) together with Lemma 2.2, there is u ∈ H₀¹(Ω) s.t., up to a subsequence, (i) {u_n}_n∈N converges to u in L^q(Ω) for all q ∈ [1,2d/(d−2)) (and weakly in L^2d/(d−2)(Ω) if d > 2) and (ii) {∇_Dnu_n}_n∈N weakly converges to ∇u in [L²(Ω)]^d. It only remains to prove that u=u. Let ϕ∈D. Owing to (2.7) together with (P2),

∇_Dn(u_n−ϕ_Tn)²_[L2(Ω)]^d≤dun−ϕ_Tn²_Tn,I ≤ d

γ₁a_Tn(u_n−ϕ_Tn, u_n−ϕ_Tn) = d

γ₁(T₁+T₂), (2.12) where T₁^def=

Ωf(x)(u_n−ϕ_Tn)(x) dxand T₂^def= a_Tn(ϕ_Tn, ϕ_Tn−u_n). Clearly, by the integrability assumptions onf and the weak convergence of{un}_n∈NtouinL^q(Ω) for allq <+∞ifd= 2 and forq=_d−2^2d ifd >2,

T₁→

Ω

f(x)(u−ϕ)(x) dxas n→ ∞. (2.13) Owing to (2.10),

a_Tn(ϕ_Tn, u_n) =

K∈Tn

σ∈EK

d_K,σ

m_σ F_K,σ(ϕ_Tn)−m_σΛ∇ϕK·n_K,σ m_σ

d_K,σ(γ_σu_n−u_n,K) +

Ω

Λ(x)∇ϕ(x)·∇Dnu_n(x) dx^def= T_2,1+T_2,2.

Using Cauchy-Schwarz inequality we conclude that T_2,1 ≤ _Dn(ϕ)un_Tn,I. Thanks to Lemma 2.3, un_Tn,I

is bounded uniformly with respect to n. Thus, by property (P3), T_2,1 → 0 asn→ ∞. Also, using the weak convergence of{∇Dnu_n}n∈N, we conclude thatT_2,2→

ΩΛ(x)∇ϕ(x)·∇u(x) dxasn→ ∞. Let us now consider the remaining terms inT₂. By Proposition2.1, ϕTnTn,I is uniformly bounded with respect to n; sinceϕ_Tn obviously converges to ϕ, it is then easy, using Lemma2.2, to see that∇_Dnϕ_Tn weakly converges to∇ϕ. Pro- ceeding in a similar way as for a_Tn(ϕ_Tn, u_n), we can thus prove thata_Tn(ϕ_Tn, ϕ_Tn)→

ΩΛ(x)∇ϕ(x)·∇ϕ(x) dx as n→ ∞. Therefore,

T₂→

Ω

Λ(x)∇ϕ(x)·∇(ϕ−u)(x) dxas n→ ∞. (2.14) Plugging (2.13) and (2.14) into the right hand side of (2.12) and using the weak convergence of∇Dn(u_n−ϕ_Tn), we conclude that, for allϕ∈D,

∇(u−ϕ)²_[L2(Ω)]^d ≤ d γ₁

Ω

f(x)(u−ϕ)(x) dx+

Ω

Λ(x)∇ϕ(x)·∇(ϕ−u)(x) dx

.

By virtue of (P1), we can apply this inequality to a sequence{ϕm}m∈N∈D which tends touinH₀¹(Ω) and let m→ ∞; sinceusolves problem (2.1), we obtain

∇(u−u)²_[L2(Ω)]^d≤ d γ₁

Ω

f(x)(u(x) −u(x)) dx−

Ω

Λ(x)∇u(x)·∇(u−u)(x) dx

= 0,

i.e.,u=u. Due to the uniqueness of the solution of (2.1), we deduce that the entire sequence{un}n∈Nconverges touinL^q(Ω) for allq∈[1,2d/(d−2)) (and weakly inL^2d/(d−2)(Ω) ifd >2). Observe that the order in which the limits for n → ∞ and m → ∞ are taken cannot be exchanged, the sequence {(ϕm)_TnTn,I}_m∈N being

possibly unbounded. This concludes the proof.

2.3. A strongly convergent gradient reconstruction

In practical applications, it is often desirable to dispose of a gradient reconstruction whichstronglyconverges to the gradient of the continuous solution (whereas ∇_D only provides a weakly convergent approximation of this gradient). Such a reconstruction can be obtained using the same formula as in the Mixed Finite Volume method of Droniou and Eymard [19].

Letσ∈ E be a mesh face, and denote byx_σ its barycenter. For allv∈H_T(Ω), define∇_Dv∈[H_T(Ω)]^d s.t., for allK∈ T,

∇_Dv(x)_|K = 1

m_KΛ⁻¹_K

σ∈EK

F_K,σ(v)(x_σ−x_K). (2.15)

For all v ∈H_T and for all K ∈ T, (∇Dv)_K will denote the (constant) value of ∇Dv onK, i.e., ∇Dv_|K(x) = (∇Dv)_K for allx∈K. The following geometrical relation holds:

∀ξ∈R^d, ∀K∈ T,

σ∈EK

m_σξ·n_K,σ(x_σ−x_K) = m_Kξ. (2.16)

Lemma 2.4 (consistency). Let {Dn}_n∈N denote a family of discretizations matching Assumption 2.1and s.t.

h_Dn→0 asn→ ∞. Then, for allϕ∈D,

n→∞lim ∇Dnϕ_Tn− ∇ϕ²_[L2(Ω)]^d = 0.

Proof. For alln∈N, for allK∈ T_n, formula (2.16) applied toξ=Λ∇ϕ_K yields, for allK∈ T, Λ⁻¹_K Λ∇ϕK= 1

m_KΛ⁻¹_K

σ∈EK

m_σΛ∇ϕK·n_K,σ(x_σ−x_K).

Let T_K,σ(ϕ_T)^def= F_K,σ(ϕ_T)−m_σΛ∇ϕK·n_K,σ. Using Cauchy-Schwarz inequality we get, for allK ∈ T and for ally∈K,

|(∇Dnϕ_Tn)_K− ∇ϕ(y)|=

1

m_KΛ⁻¹_K

σ∈EK

T_K,σ(ϕ_T)(x_σ−x_K) + 1 m_KΛ⁻¹_K

K

Λ(x)(∇ϕ(x)− ∇ϕ(y)) dx

≤ 1 α₀m_K

σ∈EK

d_K,σ

m_σ |T_K,σ(ϕ_Tn)||x_σ−x_K| d_K,σ

√m_σ+C₃β₀m_Kdiam(K)

≤ 1 α₀m_K

σ∈EK

d_K,σ

m_σ |TK,σ(ϕ_Tn)|^{2 1}₂

σ∈EK

|x_σ−x_K|² d_K,σ m_σ

¹₂

+C₃β₀

α₀ diam(K), whereC₃^def= sup_{x∈Ωi, i=1...NΩ}|ϕ(x)|. As a consequence, using (2.2) together with (2.3),

K

(∇Dnϕ_Tn)_K− ∇ϕ(y)² dy≤ 2 (α₀₁)²

σ∈EK

d_K,σ

m_σ |TK,σ(ϕ_Tn)|²× 1 m_K

σ∈EK

m_σd_K,σ

+ 2 C₃β₀

α_{0 2}

m_Kdiam(K)²

≤ 2d (α₀₁)²

σ∈EK

d_K,σ

m_σ |TK,σ(ϕ_Tn)|²+ 2 C₃β₀

α_{0 2}

m_Kh²_Dn.

Finally, summing overK∈ Tn,

∇Dnϕ_Tn− ∇ϕ²_[L2(Ω)]^d≤ 2d (α₀₁)²

2Dn(ϕ) + 2 C₃β₀

α_{0 2}

m_Ωh²_D

n.

Use (P3) to conclude.

The convergence of the gradient reconstruction (2.15) requires the following:

Assumption 2.2. Assume that there isC₄>0s.t.

∀n∈N,∀v∈H_Tn(Ω),

K∈Tn

σ∈EK

d_K,σ

m_σ |FK,σ(v)|²≤C₄v²_Tn,I. (2.17) Remark 2.5. Assumption2.2is readily verified on Λ-orthogonal meshes and two point fluxes. A mesh is said to be Λ-orthogonal if, for allσ∈ E, there existsx_σ ∈σs.t., for allK∈ Tσ, Λ⁻¹_K (x_σ−xK) is orthogonal toσ. For Λ-orthogonal meshes, we can define two-points consistent fluxes in the sense of (2.9) asF_K,σ(v) = m_σt_σ^(γσ_d^v−vK)

K,σ , where the reals{tσ}_σ∈E are given by

⎧⎨

⎩

d_K,σ

Λ_Kn_K,σ·n_K,σ + d_L,σ

Λ_Ln_L,σ·n_L,σ =d_K,σ+d_L,σ

t_σ ifσ∈ Eint with Tσ={K, L}

t_σ= Λ_Kn_K,σ·n_K,σ ifσ∈ Eext withTσ ={K}.

Since for all σ ∈ E, t_σ≤β₀, (2.17) holds with C₄ = β₀². Inequality (2.17) can be interpreted as a stability requirement on the numerical fluxes. In particular, for the G-method of Section 3, Assumption 2.2 is proved for general meshes in Lemma3.5below.

Proposition 2.2 (stability of the gradient reconstruction). Let {Dn}_n∈N denote a family of discretizations matching Definition2.1. Let Assumption 2.2hold. Then,

∀v∈H_T(Ω), ∇Dv_[L²_(Ω)]^d≤

√dC₄ α₀₁ vT,I. Proof. Letv∈H_T(Ω). Cauchy-Schwarz inequality yields

∇_Dv²_[L2(Ω)]^d=

K∈T

m_K

1

m_KΛ⁻¹_K

σ∈EK

F_K,σ(v)(x_σ−x_K)

2

≤ 1 α²₀

K∈T

σ∈EK

d_K,σ

m_σ |FK,σ(v)|²×

σ∈EK

m_σdiam(K)² d_K,σm_K

·

Owing to (2.3) together with (2.2),

σ∈EK

mσdiam(K)² dK,σmK ≤ ^d2

1. Conclude using Assumption2.2.

Theorem 2.2 (strong convergence of the discrete gradient reconstruction). Let ube the solution to (2.1). Let {Dn}n∈N be a family of meshes matching Definition 2.1 and s.t. h_Dn → 0 as n → ∞, and denote by u_n the solution of problem (2.5)on Dn. Finally, let Assumptions 2.1 and 2.2hold. Then, the sequence {∇_Dnu_n}_n∈N strongly converges to ∇uin[L²(Ω)]^d.

Proof. Thanks to Theorem2.1together with Lemma2.2, (i) the sequence of weak solutions{un}_n∈Nconverges touinL^q(Ω), for allq∈[1,2d/(d−2)) and (ii) the sequence{∇Dnu_n}n∈Nweakly converges to∇uin [L²(Ω)]^d.

Letϕ∈D. For all n∈N,

Ω

|∇_Dnu_n− ∇u|²≤3

Ω

|∇_Dnu_n− ∇_Dnϕ_Tn|²+

Ω

|∇_Dnϕ_Tn− ∇ϕ|²+

Ω

|∇ϕ(x)− ∇u|²

.

LetT_i,i= 1. . .3 denote the addends in the right hand side. Owing to Proposition 2.2together with (P2) we have

T₁≤ dC₄

(α₀₁)²γ₁ a_Tn(u_n−ϕ_Tn, u_n−ϕ_Tn).

Using the same arguments as in the proof of Theorem2.1, we infer that lim sup

n→∞ T₁≤ dC₄ (α₀₁)²γ₁

Ω

f(u−ϕ) +

Ω

Λ∇ϕ·(∇ϕ− ∇u)

.

Moreover, owing to Lemma2.4, lim_n→∞T₂= 0 and thus lim sup

n→∞

Ω

|∇_Dnu− ∇u|²≤3 dC₄ (α₀₁)²γ₁

Ω

f(u−ϕ) +

Ω

Λ∇ϕ·(∇ϕ− ∇u)

+ 3

Ω

|∇ϕ− ∇u|²

.

Finally, sinceDis dense inH₀¹(Ω), we conclude by lettingϕtend touinH₀¹(Ω) as in the proof of Theorem2.1.

3. The G method

In this section we introduce a family of FV methods generalizing the MPFA L method of [3,5] and show that it fits in the abstract analysis framework of Section2.

3.1. Construction of group gradients

The following definition collects the additional requirements on the discretization family {Dn}_n∈N for the G-method to be applicable.

Definition 3.1. Let{Dn}n∈Nbe a family of meshes matching Definition2.1. We further suppose that:

(i) Vn is a family of points (the vertices or nodes), s.t., for all K ∈ Tn, for all U_K ⊆ EK satisfying card(U_K)≥d, we have

σ∈UKσ=∅ or

σ∈UKσ=s∈ Vn. For all s ∈ Vn, we let Es def

= {σ ∈ E |s ∈ σ}

and Tsdef

= {K∈ T |s∈K}, and we assume that each σ contains at least one vertex (this could be false in dimensiond= 3 if, for example,σis only a portion of a planar face of a cell);

(ii) the number of faces sharing one node remains bounded as the mesh is refined,i.e., there exists₃ s.t.

sup

n∈Nmax

s∈Vn

card(Es)≤₃; (iii) Gis the finite family offace groups defined as follows:

G^def= {G⊂ EK∩ Es, K ∈ Ts, s∈ Vn, card(G) =d}.

For eachG∈G, we letTG def

= {K∈ T |G∩ EK =∅}. We also arbitrarily select a cell, which we denote byK_G, s.t.G⊂ EKG (see Fig.2).

Remark 3.1. In many cases, a given group Gis contained in a unique EK but, in some cases (especially if the discretization has non-convex cells), there can be multiple possible choices for K_G. In [3,5] the cellK_G is referred to as theprimary cell.

σ σ σ

σ

K_G

K_G K_G

K_G

Figure 2. Groups containing a faceσand corresponding primary cellsK_G ford= 2.

Letv∈H_T(Ω),G∈G,σ∈GandK∈ Tσ. In order to define a discrete fluxF_K,σ(v) throughσ, we construct a group gradient (∇Dv)^G,σ_K ∈R^d. Thefull gradient used in the definition ofF_K,σ(v) will then be obtained as a convex combination of the group gradients corresponding to the groupsG∈ G s.t.σ∈G(see Eq. (3.6) below).

Let us detail the procedure to obtain the group gradient (∇_Dv)^G,σ_K . First, for allσ∈ E,Tσ ={K, L}, we require that the values v_K, v_L and the gradient reconstruction (∇Dv)^G,σ_K , (∇Dv)^G,σ_L yield the same value trace on σ, i.e.,

v_K+ (∇Dv)^G,σ_K ·(x−x_K) =v_L+ (∇Dv)^G,σ_L ·(x−x_L) ∀x∈σ.

For a boundary faceσ, we require that the value obtained at its barycenterx_σ be zero. Second, we require the conservativity of the resulting fluxes,i.e.,

Λ_K(∇Dv)^G,σ_K ·n_K,σ+ Λ_L(∇Dv)^G,σ_L ·n_L,σ= 0.

The above set of equations are not sufficient to uniquely define the group gradients (and thus to estimate them, which is fundamental in the study of the numerical method). We therefore add another constraint, giving a particular role to the cell K_G selected for the group G, namely we require that (∇_Dv)^G,σ_K

G do not depend on σ∈ G, and we denote by (∇_Dv)^G_K

G the common value of this group gradient for all σ ∈ G. The discrete gradients are thus defined, as the solution of the following local problems: For allG∈Gand all σ∈G∩ Eint, withTσ ={KG, L},

v_KG+ (∇_Dv)^G_K

G·(x−x_KG) =v_L+ (∇_Dv)^G,σ_L ·(x−x_L) ∀x∈σ, Λ_KG(∇Dv)^G_K

G·n_KG,σ+ Λ_L(∇Dv)^G,σ_L ·n_L,σ= 0, (3.1) and for allσ∈G∩ Eext,

v_KG+ (∇Dv)^G_K

G·(x_σ−x_KG) = 0. (3.2)

Lemma 3.1. For each groupG ∈ G, the gradient reconstruction (∇Dv)^G_K

G defined by (3.1) and (3.2) can be obtained solving a linear system of the form

AGX_G=BG(v), (3.3)