A Finite Element Framework for Some Mimetic Finite Difference Discretizations

TL;DR

This paper establishes a connection between mimetic finite difference schemes and finite element methods, enabling better analysis and the development of efficient multigrid solvers for problems involving curl and divergence operators.

Contribution

It introduces a finite element framework for mimetic finite difference schemes, facilitating analysis and multigrid method design using finite element theory.

Findings

Equivalence relations between mimetic schemes and finite element methods.

Development of multigrid methods with demonstrated efficiency.

Use of Local Fourier Analysis and numerical tests to validate results.

Abstract

In this work we derive equivalence relations between mimetic finite difference schemes on simplicial grids and modified N\'ed\'elec-Raviart-Thomas finite element methods for model problems in $\mathbf{H}(\operatorname{\mathbf{curl}})$ and $H(\operatorname{div})$. This provides a simple and transparent way to analyze such mimetic finite difference discretizations using the well-known results from finite element theory. The finite element framework that we develop is also crucial for the design of efficient multigrid methods for mimetic finite difference discretizations, since it allows us to use canonical inter-grid transfer operators arising from the finite element framework. We provide special Local Fourier Analysis and numerical results to demonstrate the efficiency of such multigrid methods.