HDG and CG methods for the Indefinite Time-Harmonic Maxwell's Equations under minimal regularity

TL;DR

This paper introduces a hybrid HDG and CG method for Maxwell's equations with piecewise smooth coefficients, providing optimal convergence analysis and efficient Lagrange multiplier approximation.

Contribution

It extends HDG methods to piecewise smooth coefficients and employs CG for Lagrange multipliers, reducing degrees of freedom and enabling decoupled systems.

Findings

Optimal convergence estimates derived for piecewise smooth coefficients.

Efficient Lagrange multiplier approximation with fewer degrees of freedom.

Numerical experiments confirm theoretical results.

Abstract

We propose to use a hybridizable discontinuous Galerkin (HDG) method combined with the continuous Galerkin (CG) method to approximate Maxwell's equations. We make two contributions in this paper. First, even though there are many papers using HDG methods to approximate Maxwell's equations, to our knowledge they all assume that the coefficients are smooth (or constant). Here, we derive optimal convergence estimates for our HGD-CG approximation when the electromagnetic coefficients are {\em piecewise} smooth. This requires new techniques of analysis. Second, we use CG elements to approximate the Lagrange multiplier used to enforce the divergence condition and we obtain a discrete system in which we can decouple the discrete the Lagrange multiplier. Because we are using a continuous Lagrange multiplier space, the number of degrees of freedom devoted to this are less than for other HDG methods. We present numerical experiments to confirm our theoretical results.