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A symmetric discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell-centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Convergence of the approximate solution to the continuous solution is proved for general (possibly discontinuous) tensors, general (possibly nonconforming) meshes, and with no regularity assumption on the solution. An error estimate is then deduced under suitable regularity assumptions on the solution.

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We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

Gradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformity and compactness, are shown in this paper to be sufficient to prove the convergence of gradient schemes for linear and nonlinear elliptic and parabolic problems, including the case of nonlocal operators arising for example in image processing. We also show that the schemes of the Hybrid Mimetic Mixed family, which include in particular the Mimetic Finite Difference schemes, may be seen as gradient schemes meeting these four properties, and therefore converges for the class of above-mentioned problems.

Abstract. In this paper, we study some discretization schemes for diffusive flows in heterogeneous anisotropic porous media. We first introduce the notion of gradient scheme, and show that several existing schemes fall into this framework. Then, we construct two new gradient schemes which have the advantage of a small stencil. Numerical results obtained for real reservoir meshes show the efficiency of the new schemes, compared to existing ones.

One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u t +div(qf (u))−∆ϕ(u) = 0 by a piecewise constant function u D using a discretization D in space and time and a finite volume scheme. The convergence of u D to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on u D are used to prove the convergence, up to a subsequence, of u D to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of u D to u. Some numerical results on a model equation are shown. Mathematics Subject Classification: 65M12 The nonlinear parabolic degenerate problemLet Ω be a bounded open subset of R d , (d = 1, 2 or 3) with boundary ∂Ω and let T ∈ R * + . One considers the following problem.The initial condition is formulated as follows:u(x, 0) = u 0 (x) for x ∈ Ω.(2) Correspondence to: R. Eymard

This paper is devoted to the study of an error estimate of the nite volume approximation to the solution u 2 L 1 (IR N IR) of the equation ut + div(vf(u)) = 0, where v is a vector function depending on time and space. A \h 1=4 " error estimate for an initial value in BV (IR N) is shown for a large variety of nite volume monotoneous ux schemes, with an explicit or implicit time discretization. For this purpose, the error estimate is given for the general setting of approximate entropy solutions, where the error is expressed in terms of measures in IR N and IR N IR. The study of the implicit schemes involves the study of the existence and uniqueness of the approximate solution. The cases where an \h 1=2 " error estimate can be achieved are also discussed.