Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator

TL;DR

This paper analyzes a hybridizable discontinuous Galerkin (HDG) method for Maxwell equations, demonstrating stability and optimal convergence for various regularity levels through theoretical analysis and numerical experiments.

Contribution

It introduces a stable HDG method for Maxwell equations with error analysis applicable to both high and low regularity cases, including special boundary data treatment.

Findings

The HDG method is stable and converges optimally.

Numerical experiments verify theoretical convergence rates.

The approach handles low regularity boundary data effectively.

Abstract

In this paper, we study a hybridizable discontinuous Galerkin (HDG) method for the Maxwell operator. The only global unknowns are defined on the inter-element boundaries, and the numerical solutions are obtained by using discontinuous polynomial approximations. The error analysis is based on a mixed curl-curl formulation for the Maxwell equations. Theoretical results are obtained under a more general regularity requirement. In particular for the low regularity case, special treatment is applied to approximate data on the boundary. The HDG method is shown to be stable and convergence in an optimal order for both high and low regularity cases. Numerical experiments with both smooth and singular analytical solutions are performed to verify the theoretical results.