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A hybridizable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations

Gang Chen, Haijun Wu, Liwei Xu

Abstract

In this paper, we aim to develop a hybridizable discontinuous Galerkin (HDG) method for the indefinite time-harmonic Maxwell equations with the perfectly conducting boundary in the three-dimensional space. First, we derive the wavenumber explicit regularity result, which plays an important role in the error analysis for the HDG method. Second, we prove a discrete inf-sup condition which holds for all positive mesh size $h$, for all wavenumber $k$, and for general domain $Ω$. Then, we establish the optimal order error estimates of the underlying HDG method with constant independent of the wavenumber. The theoretical results are confirmed by numerical experiments.

A hybridizable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations

Abstract

In this paper, we aim to develop a hybridizable discontinuous Galerkin (HDG) method for the indefinite time-harmonic Maxwell equations with the perfectly conducting boundary in the three-dimensional space. First, we derive the wavenumber explicit regularity result, which plays an important role in the error analysis for the HDG method. Second, we prove a discrete inf-sup condition which holds for all positive mesh size , for all wavenumber , and for general domain . Then, we establish the optimal order error estimates of the underlying HDG method with constant independent of the wavenumber. The theoretical results are confirmed by numerical experiments.

Paper Structure

This paper contains 9 sections, 23 theorems, 112 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The wavenumber explicit regularity
  3. An HDG method
  4. Elliptic projection
  5. Approximation errors
  6. Discrete inf-sup condition
  7. Error estimates of the elliptic projection
  8. Error estimates of the HDG methods
  9. Numerical experiments

Key Result

Lemma 2.1

For any $\bm{v}\in\bm L^2(\Omega)$, there exist functions $\bm z\in \bm H^1(\Omega)$ and $\psi\in H^1_0(\Omega)$ such that and

Theorems & Definitions (38)

  • Lemma 2.1: Helmholtz decomposition MR2059447
  • Lemma 2.2: cf. Maxwell-1997
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 28 more