A nonconforming Trefftz virtual element method for the Helmholtz problem: numerical aspects

TL;DR

This paper presents an improved nonconforming Trefftz virtual element method for the 2D Helmholtz problem, focusing on reducing ill-conditioning and demonstrating robustness through extensive numerical experiments including acoustic scattering.

Contribution

It introduces an automatic basis filtering strategy to reduce degrees of freedom and ill-conditioning, enhancing the method's robustness and efficiency.

Findings

Significant reduction in ill-conditioning through basis filtering.

Effective performance in acoustic scattering applications.

Robust results across h-, p-, and hp-versions of the method.

Abstract

We discuss the implementation details and the numerical performance of the recently introduced nonconforming Trefftz virtual element method for the 2D Helmholtz problem. In particular, we present a strategy to significantly reduce the ill-conditioning of the original method; such a recipe is based on an automatic filtering of the basis functions edge by edge, and therefore allows for a notable reduction of the number of degrees of freedom. A widespread set of numerical experiments, including an application to acoustic scattering, the $h$-, $p$-, and $hp$-versions of the method, is presented. Moreover, a comparison with other Trefftz-based methods for the Helmholtz problem shows that this novel approach results in robust and effective performance.