Convergence analysis of the mimetic finite difference method for elliptic problems with staggered discretization of the diffusion coefficients

TL;DR

This paper analyzes the convergence of a new family of mimetic finite difference schemes for elliptic diffusion problems, featuring a staggered discretization of diffusion coefficients that enhances flexibility and preserves key mathematical properties.

Contribution

It introduces a novel staggered discretization approach for mimetic schemes, ensuring stability, convergence, and compatibility with efficient solvers for elliptic problems.

Findings

Schemes are inf-sup stable

A priori error estimates are established

Numerical examples confirm convergence and accuracy

Abstract

We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic operator, i.e., the discrete divergence, and the inner product in the space of gradients. The diffusion coefficient is therefore evaluated on different mesh locations, i.e., inside mesh cells and on mesh faces. Such a staggered discretization may provide the exibility necessary for future development of efficient numerical schemes for nonlinear problems, especially for problems with degenerate coefficients. These new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem, which allow us to use efficient algebraic solvers such as the preconditioned Conjugate Gradient method. We show that these schemes are inf-sup stable and establish a priori error estimates for the approximation of the scalar and vector solution fields. Numerical examples confirm the convergence analysis and the effectiveness of the method in providing accurate approximations.