Adaptive Refinement for $hp$-Version Trefftz Discontinuous Galerkin Methods for the Homogeneous Helmholtz Problem

TL;DR

This paper introduces an $hp$-adaptive refinement strategy for Trefftz discontinuous Galerkin methods, combining mesh, basis, and directional adaptivity to efficiently solve the homogeneous Helmholtz problem.

Contribution

It presents a novel adaptive refinement procedure that integrates mesh subdivision, basis enrichment, and directional alignment for improved Helmholtz problem solutions.

Findings

Efficient $hp$-adaptive method demonstrated through numerical experiments.

Directional adaptivity improves solution accuracy.

Empirical a posteriori error indicator guides refinement effectively.

Abstract

In this article we develop an $hp$-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h-refinement) and local basis enrichment (p-refinement), but also incorporates local directional adaptivity, whereby the elementwise plane wave basis is aligned with the dominant scattering direction. Numerical experiments based on employing an empirical a posteriori error indicator clearly highlight the efficiency of the proposed approach for various examples.