Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties

TL;DR

This paper introduces three families of quasi-Trefftz functions for the 3D convected Helmholtz equation, focusing on their construction and approximation capabilities, with a particular emphasis on a polynomial basis that avoids ill-conditioning.

Contribution

It defines, constructs, and analyzes three types of quasi-Trefftz functions for the convected Helmholtz equation, highlighting the polynomial basis's advantages over wave-based methods.

Findings

The polynomial basis avoids ill-conditioning issues.

Two wave-based families generalize plane wave solutions.

The methods show promising approximation properties.

Abstract

Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions but rather local approximate solutions to the governing PDE. In order to develop quassi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation, the present work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial. The polynomial basis shows significant promise as it does not suffer from the ill-conditioning issue inherent to wave-like bases.