Discontinuous Galerkin Methods with Trefftz Approximation

TL;DR

This paper introduces a novel space-time Trefftz-based Discontinuous Galerkin method for wave propagation that achieves spectral convergence and inherently incorporates high order time integration.

Contribution

It develops a high-order, spectral convergent DG method using Trefftz basis functions that exactly satisfy PDEs and boundary conditions element-wise.

Findings

Spectral convergence in the L2 norm demonstrated.

High order time integration is inherently achieved.

Method outperforms traditional space-only approximation methods.

Abstract

We present a novel Discontinuous Galerkin Finite Element Method for wave propagation problems. The method employs space-time Trefftz-type basis functions that satisfy the underlying partial differential equations and the respective interface boundary conditions exactly in an element-wise fashion. The basis functions can be of arbitrary high order, and we demonstrate spectral convergence in the $\Lebesgue_2$-norm. In this context, spectral convergence is obtained with respect to the approximation error in the entire space-time domain of interest, i.e. in space and time simultaneously. Formulating the approximation in terms of a space-time Trefftz basis makes high order time integration an inherent property of the method and clearly sets it apart from methods, that employ a high order approximation in space only.