Alternative basis functions for nonpolynomial finite element methods for wave problems

Much progress has been made over the last years in the development of nonpolynomial finite element methods for wave problems. The idea of these methods is to use basis functions in each element that are better adapted to the underlying PDE than polynomials. Most implementations of nonpolynomial finite element methods use local plane wave basis sets. But other PDE adapted basis sets are possible, e.g. fundamental solutions or Fourier-Bessel functions. In this talk we compare different basis sets, analyse their stability and convergence properties and show how these results can be used to design an efficient nonpolynomial finite element method for scattering on polygons.