An Embedded Trefftz DG Approach to Least-Squares Finite Element Methods

Tim van Beeck (University of Göttingen) is going to talk about: "An Embedded Trefftz DG Approach to Least-Squares Finite Element Methods"

Least-squares finite element methods (LSFEM) minimise the residual of the PDE directly. This gives discrete problems that are stable by construction, have symmetric positive definite system matrices, and come with a built-in reliable and efficient error estimator. In this talk, we present an embedded Trefftz DG approach to LSFEM. Trefftz methods use basis functions that locally lie in the kernel of the differential operator, which greatly reduces the number of degrees of freedom. The embedded Trefftz DG method obtains these spaces through an element-local embedding into a standard DG space, so no explicit Trefftz basis is needed. We apply the method to a range of linear second-order problems, including Helmholtz-type equations and PDEs in non-divergence form, and show that it keeps the properties above under suitable mesh resolution conditions, while requiring far fewer degrees of freedom than standard DG or first-order least-squares formulations. We conclude with an outlook on nonlinear problems such as the Hamilton-Jacobi-Bellman equation.