Trefftz Discontinuous Galerkin methods for scattering by periodic structures

TL;DR

This paper introduces a Trefftz discontinuous Galerkin method for simulating wave scattering by periodic structures, utilizing plane wave spaces and a new stability estimate for the Helmholtz equation.

Contribution

It develops a novel TDG method for periodic diffraction problems, including explicit stability analysis and analytical computation of system entries.

Findings

Stable in the small material jump limit

Analytical computation of linear-system entries

Effective for polygonal meshes

Abstract

We propose a Trefftz discontinuous Galerkin (TDG) method for the approximation of plane wave scattering by periodic diffraction gratings, modelled by the two-dimensional Helmholtz equation. The periodic obstacle may include penetrable and impenetrable regions. The TDG method requires the approximation of the Dirichlet-to-Neumann (DtN) operator on the periodic cell faces, and relies on plane wave discrete spaces. For polygonal meshes, all linear-system entries can be computed analytically. Using a Rellich identity, we prove a new explicit stability estimate for the Helmholtz solution, which is robust in the small material jump limit.