Finite Element Methods for Maxwell's Equations

TL;DR

This paper surveys finite element methods for Maxwell's equations, comparing error estimates for different approaches, highlighting the advantages of Discontinuous Galerkin methods despite less developed error analysis.

Contribution

It provides a comprehensive comparison of error estimates for conforming and DG finite element methods applied to Maxwell's equations, emphasizing DG methods' potential.

Findings

Error estimates for conforming methods with smooth coefficients

Less advanced error analysis for DG methods

DG methods show significant potential advantages

Abstract

We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.