Advanced applications and analysis of Trefftz methods

Partial differential equations (PDEs) are mathematical equations that describe the relationships between the partial derivatives of multivariable functions and are essential in modeling various physical, chemical, and biological phenomena. Numerical methods are computational techniques, and are employed to approximate solutions to PDEs, particularly when analytical solutions are difficult or impossible to obtain. The finite element method (FEM) is one such technique, discretizing the problem domain into smaller elements to facilitate the approximation using piecewise functions with localized influence. Trefftz methods can be used to reduce the computational cost of numerical methods for PDEs by constructing specific functions for the approximation. In recent years, Trefftz basis functions have been used mainly in combination with discontinuous Galerkin (DG) methods. DG methods are FEMs that allow for discontinuities in the discrete functions of a FEM, and are therefore well-suited to accommodate non-standard basis functions. Trefftz methods are typically limited to PDE problems with zero right hand side, constant coefficients, and usually require the explicit construction of Trefftz basis functions. We aim to expand the capabilities of the Trefftz-DG method, making it a versatile tool for reducing degrees of freedom in a wide range of numerical problems. Through this project, we will provide analytical tools for advanced applications and analysis of Trefftz methods. Our research will establish connections from Trefftz methods to adjacent topics, such as unfitted finite elements, problems in heterogeneous media, and fluid dynamic applications. Key techniques for the derivation of new methods are the quasi-Trefftz and the embedded Trefftz method, which allow to remove restrictions of the standard Trefftz method and circumvent explicit construction of Trefftz basis functions.