Higher Order BEM-Based FEM on Polygonal Meshes

Abstract

The BEM-based finite element method is reviewed and extended with higher order basis functions on general polygonal meshes. These functions are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are treated by means of boundary integral formulations and are approximated using the boundary element method in the numerics. To obtain higher order convergence, a new approximation of the material coefficient is proposed since previous strategies are not sufficient. Following recent ideas, error estimates are proved which guarantee quadratic convergence in the $H^1$-norm and cubic convergence in the $L_2$-norm. The numerical realization is discussed and several experiments confirm the theoretical results.