Quasi-Trefftz methods for time-harmonic wave propagation and beyond

In the field on numerical partial differential equations (PDEs), many methods rely on standard bases, like polynomial bases, but other methods rely on problem-dependent bases. Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs via Galerkin-type formulations discretized with basis functions solving exactly the governing PDE locally, making explicit use of information about the ambient medium. They are of particular interest for wave propagation problems, for example with plane wave bases with the wave number depending on the propagation medium. For context, the presentation will start with an introduction to Trefftz methods.

However, in general, for problems modeled by PDEs with variable coefficients, no exact solutions are available. Hence quasi-Trefftz methods have been introduced to address this problem: they rely not on exact solutions to the PDE but instead on high order approximate solutions constructed locally. We will discuss some of the fundamental properties of these numerical methods as well as some recent advances.
 

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