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

TL;DR

This paper demonstrates that Mimetic Finite Difference, Hybrid Finite Volume, and Mixed Finite Volume methods are essentially equivalent, enabling the transfer of theoretical results and connecting these methods to other finite element schemes.

Contribution

It unifies several discretization methods for anisotropic diffusion, revealing their equivalence and extending mathematical properties across a common framework.

Findings

Methods are identical up to generalizations

Mathematical results extend across the unified framework

For isotropic operators on specific meshes, the method simplifies to a known scheme

Abstract

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.