Generalized Plane Wave quasi-Trefftz spaces for wave propagation in inhomogeneous media

TL;DR

This paper introduces generalized plane wave quasi-Trefftz spaces for wave propagation in inhomogeneous media, offering a novel basis construction that enhances numerical solutions of variable-coefficient PDEs.

Contribution

It develops a new class of basis functions called generalized plane waves, extending quasi-Trefftz methods to inhomogeneous media with variable coefficients.

Findings

New basis functions for wave PDEs in inhomogeneous media

Connection between generalized plane waves and polynomial quasi-Trefftz bases

Potential improvements in numerical wave propagation simulations

Abstract

Partial Differential Equations (PDEs) models for wave propagation in inhomogeneous media are relevant for many applications. We will discuss numerical methods tailored for tackling problems governed by these variable-coefficient PDEs. Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs via Galerkin methods using basis functions that are exact solutions of the PDE, making explicit use of information about the ambient medium. However, wave propagation in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available. Quasi-Trefftz methods have been introduced, in the case of the Helmholtz equation, to address this problem: they rely instead on high-order approximate solutions constructed locally. We will discuss basis of Generalized Plane Waves, a particular kind of quasi-Trefftz functions, and how their construction can be related to the construction of polynomial quasi-Trefftz bases.