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

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

A mixed finite volume scheme for anisotropic diffusion problems on any grid

We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. In the case of simplicial meshes, the approximate solution is shown to converge to the continuous ones as the size of the mesh tends to 0, and an error estimate is given. In the general case, we propose a slightly modified scheme for which we again prove the convergence, and give an error estimate. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids

Monotone corrections for generic cell-centered Finite Volume approximations of anisotropic diffusion equations

We present a nonlinear technique to correct a general Finite Volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many Finite Volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding solutions converge to the continuous one as the size of the mesh tends to 0. Finally we present numerical results showing these corrections suppress local minima produced by the initial Finite Volume scheme

An analysis of the isoparametric bilinear finite volume element method by applying the Simpson rule to quadrilateral meshes

In this work, we construct and study a special isoparametric bilinear finite volume element scheme for solving anisotropic diffusion problems on general convex quadrilateral meshes. The new scheme is obtained by employing the Simpson rule to approximate the line integrals in the classical isoparametric bilinear finite volume element method. By using the cell analysis approach, we suggest a sufficient condition to ensure the coercivity of the new scheme. The sufficient condition has an analytic expression, which only involves the anisotropic diffusion tensor and the geometry of quadrilateral mesh. This yields that for any diffusion tensor and quadrilateral mesh, we can directly judge whether this sufficient condition is satisfied. Specifically, this condition covers the traditional $h^{1+\gamma}$-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. An optimal $H^1$ error estimate of the proposed scheme is also obtained for a quasi-parallelogram mesh. The theoretical results are verified by some numerical experiments

Finite volume scheme based on cell-vertex reconstructions for anisotropic diffusion problems with discontinuous coefficients

We propose a new second-order finite volume scheme for non-homogeneous and anisotropic diffusion problems based on cell to vertex reconstructions involving minimization of functionals to provide the coefficients of the cell to vertex mapping. The method handles complex situations such as large preconditioning number diffusion matrices and very distorted meshes. Numerical examples are provided to show the effectiveness of the method

Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

International audienceIn this paper, we prove the convergence of a discrete duality finite volume scheme for a system of partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration. We first establish some a priori estimates satisfied by the sequences of approximate solutions. Then, it yields the compactness of these sequences. Passing to the limit in the numerical scheme, we finally obtain that the limit of the sequence of approximate solutions is a weak solution to the problem under study