Some possibly degenerate elliptic problems with measure data and non linearity on the boundary

The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates

A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension

Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality condition. This discrete gradient is shown to satisfy a strong convergence property on the interpolation of regular functions, and a weak one on functions bounded for a discrete $H^1$ norm. To highlight the importance of both properties, the convergence of the finite volume scheme on a homogeneous Dirichlet problem with full diffusion matrix is proven, and an error estimate is provided. Numerical tests show the actual accuracy of the method

Finite volume schemes for two phase flow in porous media

The system of equations obtained from the conservation of multiphasic fluids in porous media is usually approximated by finite volume schemes in the oil reservoir simulation setting. The convergence properties of these schemes are only known in a few simplified cases. The aim of this paper is to present some new results of convergence in more complex cases. These results are based on an adaptation of the H-convergence notion to the limit of discrete approximates

Convergence of linear finite elements for diffusion equations with measure data

We show here the convergence of the linear finite element approximate solutions of a diffusion equation to a weak solution, with weak regularity assumptions on the data

H-convergence and numerical schemes for elliptic equations

We study the convergence of two coupled numerical schemes, which are a discretization of a so-called elliptic-hyperbolic system. Only weak convergence properties are proved on the discrete diffusion of the elliptic problem, and an adaptation of the H-convergence method gives a convergence property of the elliptic part of the scheme. The limit of the approximate solution is then the solution of an elliptic problem, the diffusion of which is not in the general case the H-limit of the discrete diffusion. In a particular case, a kind of weak limit is then obtained for the hyperbolic equation

A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

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

Compactness of discrete approximate solutions to parabolic PDEs - Application to a turbulence model

International audienceIn this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function u (n) of the sequence with respect to n. This compactness result is then used to prove the convergence of a numerical scheme combining finite volumes and finite elements for the solution of a reduced turbulence problem. 1. Introduction. Let us suppose given a sequence of approximations of a parabolic problem (P), which, for instance, may be though of as discretizations of (P) by a numerical scheme, or resulting from the combination of a Faedo-Galerkin technique with a time discretization. In both cases, we are in presence of a family of finite-dimensional systems, and their solution, let us say (u (n)) n∈N , may be considered as a sequence of functions of time, taking their values in a finite dimensional subspace, let us say B (n) of a Banach space (usually a Lebesgue L q space, q ≥ 1). To show the convergence of such a process, a common path is to follow the following steps: 1. first, for each approximate problem, prove the existence of a solution, and derive estimates satisfied by any solution, 2. then use compactness arguments to show (possibly up to the extraction of a subsequence) the existence of a limit, 3. and, finally, prove that this limit satisfies the initial problem (P). Let us now focus on item 2, which is the issue addressed in this paper. The problem here is to prove a compactness result for a sequence of functions of time taking their value in a sequence of discrete spaces and controlled (themselves and their discrete time-derivative) in discrete norms; in the general case, both B (n) and the space part of the norms depend on n. The compactness result given in this paper relies on this particular structure, and consists in a generalization of the classical Aubin-Simon lemma to this specific case

Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces

International audienceA 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. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution

A finite volume scheme for anisotropic diffusion problems

A new finite volume for the discretization of anisotropic diffusion problems on general unstructured meshes in any space dimension is presented. The convergence of the approximate solution and its discrete gradient is proven. The efficiency of the scheme is shown through numerical examples