Plane Wave Discontinuous Galerkin methods for the Helmholtz equation and Maxwell equations in Anisotropic Media

TL;DR

This paper develops and analyzes plane wave discontinuous Galerkin methods for solving Helmholtz and Maxwell equations in three-dimensional anisotropic media, providing new basis functions, error estimates, and numerical validation.

Contribution

It introduces novel plane wave basis functions tailored for anisotropic media and derives error estimates considering matrix coefficients and mesh regularity.

Findings

Numerical results confirm high accuracy of the proposed PWDG method.

Error estimates depend on the condition number of coefficient matrices.

The method is effective for complex anisotropic media problems.

Abstract

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. Then we derive the error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under a new assumption on the shape regularity of polyhedral meshes. Numerical results verify the validity of the theoretical results, and indicate that the approximate solutions generated by the proposed PWDG method possess high accuracy.