7th April – ‪Christoph Lehrenfeld: Embedded Trefftz Discontinuous Galerkin methods

Discontinuous Galerkin (DG) methods are widely used to discretize partial differential equations (PDEs) due to (a.o.) flexibility for designing robust methods and simplicity in terms of data structures. One major drawback of DG methods is, however, the increased number of (globally coupled) degrees of freedom (ndof) compared to, for instance, continuous Galerkin methods.

One – by now established – remedy is the use of Hybrid DG methods. These allow reducing the globally coupled ndof essentially by introducing the concept of static condensation for DG methods. Thereby the dimensions of global linear systems that need to be solved for reduced from O(p^d) to O(p^{d−1}), where d is the space dimension and p is the polynomial degree of the finite element space.

A different approach is the use of Trefftz DG methods, where a DG formulation is modified by restricting the finite element spaces to functions that element-wise solve the PDE at hand. This results in a similar reduction of globally coupled ndof, cf. Figure 1 below. However, due to several limitations, Trefftz DG methods have only been applied for special PDEs so far. On the one hand, the finite element spaces have to be specifically tailored for each PDE type, on the other hand, inhomogeneous equations and non-constant coefficients in the differential operators are difficult to deal with and rarely treated.

In this talk, we introduce Embedded Trefftz DG methods which exploit the existence of an underlying standard DG formulation for an efficient and flexible implementation of Trefftz DG methods. Furthermore, we relax Trefftz DG methods in view of the constraints on the finite element space leading to weak Trefftz DG spaces. Both together allow us to remove the limitations of Trefftz DG methods in order to enable them for a large range of PDE problems.