A numerical study of the dispersion and dissipation properties of virtual element methods for the Helmholtz problem

We study numerically the dispersion and dissipation properties of the plane wave virtual element method and the nonconforming Trefftz virtual element method for the Helmholtz problem. Whereas the former method is based on a conforming virtual partition of unity approach in the sense that the local (implicitly defined) basis functions are given as modulations of lowest order harmonic virtual element functions with plane waves, the latter one represents a pure Trefftz method with local edge-related basis functions that are eventually glued together in a nonconforming fashion. We will see that the qualitative and quantitative behavior of dissipation and dispersion of the method hinges upon the level of conformity and the use of Trefftz basis functions. To this purpose, we also compare the results to those obtained in [15] for the plane wave discontinuous Galerkin method, and to those for the standard polynomial based finite element method.