Solution of Helmholtz problems by knowledge-based FEM

Abstract

The numerical solution of Helmholtz' equation at large wavenumber is very expensive if attempted by "traditional" discretisation methods (FDM, standard Galerkin FEM). For reliable results, the mesh has to be very fine. The bad performance of the traditional FEM for Helmholtz problems can be related to the deterioration of stability of the Helmholtz differential operator at high wavenumber. As an alternative, several non-standard FEM have been proposed in the literature. In these methods, stabilisation is either attempted directly by modification of the differential operator or indirectly, via improvement of approximability by the incorporation of particular solutions into the trial space of the FEM. It can be shown that the increase in approximability can make up for the stability loss, thus improving significantly the convergence behavior of the knowledge based FEM compared to the standard approach. In our paper, we refer recent results on stability and convergence of h- and h-p-Galerkin ("standard") FEM for Helmholtz problems. We then review, under the label of "knowledge-based" FEM, several approaches of stabilised FEM as well as high-approximation methods like the Partition of Unity and the Trefftz method. The performance of the methods is compared on a two-dimensional model problem.