A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic electromagnetic problems

TL;DR

This paper extends a hybridizable discontinuous Galerkin method with transmission variables to solve time-harmonic Maxwell equations efficiently, demonstrating well-posedness and contraction properties, and implementing a high-performance 3D solver.

Contribution

It adapts the CHDG method from Helmholtz to Maxwell equations, proving well-posedness and contraction, and develops a parallel 3D implementation for electromagnetic problems.

Findings

The hybridized system is well-posed.

The fixed-point iteration is contractive.

Numerical examples show efficiency and scalability.

Abstract

The CHDG method is a hybridizable discontinuous Galerkin (HDG) finite element method suitable for the iterative solution of time-harmonic wave propagation problems. Hybrid unknowns corresponding to transmission variables are introduced at the element interfaces and the physical unknowns inside the elements are eliminated, resulting in a hybridized system with favorable properties for fast iterative solution. In this paper, we extend the CHDG method, initially studied for the Helmholtz equation, to the time-harmonic Maxwell equations. We prove that the local problems stemming from hybridization are well-posed and that the fixed-point iteration naturally associated to the hybridized system is contractive. We propose a 3D implementation with a discrete scheme based on nodal basis functions. The resulting solver and different iterative strategies are studied with several numerical examples using a high-performance parallel C++ code.