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

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.

Figures (6)

• Figure 1: Notation for the outgoing and incoming transmission variables at the face $F$ shared by an element $K$ and a neighboring element $K'$.
• Figure 1: Plots of $\operatorname{Re}(\boldsymbol{e}_{\mathrm{ref}})$ for the reference solutions. The coloring corresponds to the amplitude of the vector.
• Figure 2: Results of the different iterative solvers for the Benchmarks 1 (free space) and 2 (cavity).
• Figure 3: Geometry of the COBRA domain with the aperture in gray (left) and the subdomain partition with 160 subdomains (right).
• Figure 4: Snapshots of $\left|\operatorname{Re}(\boldsymbol{e}_h)\right|$ for the COBRA benchmark at four different points during the solution process using modal CGNR, cf. §\ref{['sect:cobra']}.

Theorems & Definitions (9)

• Theorem 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Theorem 3.3
• Proof 3
• Theorem 3.4
• Proof 4
• Corollary 3.5