Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements

TL;DR

This paper presents an error analysis of a mixed discontinuous Galerkin method for time-harmonic Maxwell equations, achieving optimal convergence with minimal smoothness assumptions and introducing a practical stabilization parameter.

Contribution

The paper develops a novel error analysis framework for DG methods applied to Maxwell equations with minimal regularity, including explicit stabilization parameter computation.

Findings

Achieved optimal convergence rates under minimal smoothness.

Introduced a lifting operator to handle trace integrals.

Provided an explicit stabilization parameter.

Abstract

An error analysis of a mixed discontinuous Galerkin (DG) method with Brezzi numerical flux for the time-harmonic Maxwell equations with minimal smoothness requirements is presented. The key difficulty in the error analysis for the DG method is that the tangential or normal trace of the exact solution is not well-defined on the mesh faces of the computational mesh. We overcome this difficulty by two steps. First, we employ a lifting operator to replace the integrals of the tangential/normal traces on mesh faces by volume integrals. Second, optimal convergence rates are proven by using smoothed interpolations that are well-defined for merely integrable functions. As a byproduct of our analysis, an explicit and easily computable stabilization parameter is given.