Séance Séminaire

NEMESIS Seminar

Quasi-Trefftz Spaces for Linear Differential Equations

Trefftz methods are a class of high-order numerical schemes for solving problems modeled by partial differential equations (PDEs), and have been studied extensively for wave propagation problems. They construct the numerical solution within a space of functions specifically locally tailored to the PDE considered, with test and trial functions that are solutions of the PDE under consideration on each mesh element. Notably, Trefftz methods achieve a level of accuracy with reduced computational cost compared to traditional approaches like finite elements or standard Discontinuous Galerkin methods. For problems of wave propagation through inhomogeneous media, the inhomogeneity is modeled by variable coefficients in the governing PDE. However, the application of Trefftz methods is limited to problems governed by linear, homogeneous and piecewise-constant coefficients PDEs, as exact solutions are often unavailable in more complex cases. To address this limitation, quasi-Trefftz methods use element-wise approximate solutions of the PDE, enabling the analysis of a wider range of problems while maintaining the good properties of Trefftz schemes. Recent studies have demonstrated the convergence and stability of this quasi-Trefftz approach for some scalar problems, for example for the diffusion-advection-reaction equation. After an overview of quasi-Trefftz techniques, in this talk I will present recent results and novel insights aimed at expanding the applications of quasi-Trefftz methods to complex vector-valued PDEs, including time-harmonic Maxwell's equations.