A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation

TL;DR

This paper introduces a novel Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation, providing theoretical analysis, convergence rates, and numerical validation, especially effective at high frequencies.

Contribution

It develops and analyzes a new Trefftz basis-based space-time dG method for wave equations, including convergence proofs and practical performance insights.

Findings

Proven best approximation properties for the method.

Established convergence rates in arbitrary dimensions.

Numerical experiments confirm effectiveness at high frequencies.

Abstract

A new space-time discontinuous Galerkin (dG) method utilising special Trefftz polynomial basis functions is proposed and fully analysed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behaviour of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions $d = 1$ and $d = 2$. Numerical experiments highlight the effectiveness of the Trefftz method in problems with energy at high frequencies.