Contemporary Challenges in Trefftz Methods, from Theory to Applications

Trefftz methods are a class of methods for the numerical simulation of problems modeled by differential equations. As opposed to other methods such as finite elements, relying on a unique representation of solutions for all possible equations, Trefftz methods rely on a representation of the solution that is tailored for each particular problem, that is for each differential equation. Thanks to this tailored representation, the methods enjoy desirable properties: they can approximate solutions within a given accuracy using less computational resources than for instance finite element methods. In the past few years, the mathematical community has reached significant breakthroughs to address several challenges attached to the practical performance of Trefftz methods. By gathering experts on these methods, as well as scientists from both academia and industry who use them towards various applications, we have a unique opportunity to exchange about recent developments, stimulate new interactions and shape the future developments on Trefftz methods.