Trefftz methods with evanescent plane waves

TL;DR

This paper introduces a simple method to select evanescent plane wave basis functions for Trefftz methods, significantly improving numerical stability and accuracy in Helmholtz problem solutions.

Contribution

It proposes a new basis selection recipe for Trefftz methods using evanescent waves, enhancing their stability and performance.

Findings

Numerical results show improved stability with evanescent basis functions.

The Ultraweak Variational Formulation benefits from the new basis choice.

Further details and examples are forthcoming in a future publication.

Abstract

Classical Trefftz methods approximate Helmholtz solutions using propagative plane waves and are subject to strong numerical instabilities. Evanescent plane wave bases can substantially mitigate this phenomenon. We propose a simple recipe to select such basis functions. We show that the numerical results obtained by the Ultraweak Variational Formulation (UWVF) greatly improve thanks to this choice. More details and examples will soon be available in [Galante, Moiola, Parolin 2026].