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

Approximation d'un problème biharmonique par élément fini P1

International audienceWe propose an approximation of the solution of the biharmonic problem in $H_0^2(\Omega)$ which relies on the discretization of the Laplace operator using nonconforming continuous piecewise linear finite elements.Nous proposons une approximation de la solution du problème bi-harmonique dans $H_0^2(\Omega)$ basée sur la discrétisation du Laplacien par éléments finis P1 continus mais non conformes

A staggered finite volume scheme on general meshes for the Navier-Stokes equations in two space dimensions

International audienceThis paper presents a new finite volume scheme for the incompressible steady-state Navier-Stokes equations on a general 2D mesh. Thescheme is staggered, i.e. the discrete velocities are not located at the same place as the discrete pressures. We prove the existence and the uniqueness of a discrete solution for a centered scheme under a condition on the data, and the unconditional existence of a discrete solution for an upstream weighting scheme. In both cases (nonlinear centered and upstream weighting schemes), we prove the convergence of a penalized version of the scheme to a weak solution of the problem. Numerical experiments show the efficiency of the schemes on various meshes

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

The Lions domain decomposition algorithm on non matching cell-centered finite volume meshes

A finite volume scheme for convection diffusion equation on non matching grids is presented. Sharp error estimates for $H^2$ solutions of the continuous problem are obtained. A finite volume version of an adaptation of the Schwarz algorithm due to P.L. Lions is then studied. For a fixed mesh, its convergence towards the finite volume scheme on the whole domain is proven. Numerical experiments are performed to illustrate the theoretical rate of convergence of the finite volume sequences of solutions as the mesh is refined, together with the speed of convergence of the Schwarz algorithm

Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations

International audienceGradient 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 Hybrid Mixed Mimetic family, which includes 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