Arbitrary order Trefftz-like basis functions on polygonal meshes and realization in BEM-based FEM

Polygonal meshes appear in more and more applications and the BEM-based Finite Element Method (FEM) turns out to be a forward-looking approach. The method uses Trefftz-like basis functions which are defined implicitly and are treated locally by means of Boundary Element Methods (BEMs). The BEM-based Finite Element Method is applicable on a variety of meshes including hanging nodes. The aim of this presentation is to give a rigorous construction of H1H1-conforming basis functions of a given arbitrary order yielding optimal rates of convergence in a Finite Element Method for elliptic equations. With the help of an interpolation operator, approximation properties are proven which guarantee optimal rates of convergence in the H1H1- as well as in the L2L2-norm for Finite Element simulations. These theoretical results are illustrated and verified by several numerical examples on polygonal meshes.