FEM with Trefftz trial functions on polyhedral elements

The goal of this paper is to generalize a variant of the BEM-based FEM for second order elliptic boundary value problems to three space dimensions. Here, the emphasis lies on polyhedral meshes with polygonal faces, where even non-convex elements are allowed. Due to an implicit definition of the trial functions in the spirit of Trefftz, the strategy yields conforming approximations and is very flexible with respect to the meshes. Thus, it gets into the line of recent developments in several areas. The arising local problems are treated by two dimensional Galerkin schemes coming from finite and boundary element formulations. With the help of a new interpolation operator and its properties, convergence estimates are proven in the H1- as well as in the L2-norm. Numerical experiments confirm the theoretical results.