A finite-element framework for a mimetic finite-difference discretization of Maxwell's equations

TL;DR

This paper introduces a mimetic finite-difference discretization for Maxwell's equations, demonstrating its equivalence to a finite-element scheme, which facilitates analysis and the development of efficient solvers.

Contribution

It establishes a connection between MFD and FE methods for Maxwell's equations, enabling better analysis and solver design for the discretized system.

Findings

MFD discretization preserves physical properties of Maxwell's equations.

Equivalence to FE scheme after mass-lumping and scaling.

Block preconditioners are effective for the MFD system.

Abstract

Maxwell's equations are a system of partial differential equations that govern the laws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization of the equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the MFD discretization is equivalent to a structure-preserving finite-element (FE) scheme. This allows for a transparent analysis of the MFD method using the FE framework, and provides an avenue for the construction of efficient and robust linear solvers for the discretized system. In particular, block preconditioners designed for FE formulations can be applied to the MFD system in a straightforward fashion. We present numerical tests which verify the accuracy of the MFD scheme and confirm the robustness of the preconditioners.