A Trefftz Discontinuous Galerkin Method for Time Harmonic Waves with Generalized Impedance Boundary Conditions

TL;DR

This paper introduces a Trefftz Discontinuous Galerkin method tailored for the Helmholtz equation with generalized impedance boundary conditions, effectively modeling scattering from coated or absorbing boundaries.

Contribution

It develops a coupled TDG and surface finite element scheme for GIBCs, proving convergence and demonstrating effectiveness through numerical examples.

Findings

Convergence of the proposed scheme is theoretically established.

Numerical examples validate the accuracy and applicability of the method.

The approach effectively models scattering with thin coatings or absorbing boundaries.

Abstract

We show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.