On stability of discretizations of the Helmholtz equation (extended version)

摘要

We review the stability properties of several discretizations of theHelmholtz equation at large wavenumbers. For a model problem in a polygon, acomplete $k$-explicit stability (including $k$-explicit stability of thecontinuous problem) and convergence theory for high order finite elementmethods is developed. In particular, quasi-optimality is shown for a fixednumber of degrees of freedom per wavelength if the mesh size $h$ and theapproximation order $p$ are selected such that $kh/p$ is sufficiently small and$p = O(\log k)$, and, additionally, appropriate mesh refinement is used nearthe vertices. We also review the stability properties of two classes ofnumerical schemes that use piecewise solutions of the homogeneous Helmholtzequation, namely, Least Squares methods and Discontinuous Galerkin (DG)methods. The latter includes the Ultra Weak Variational Formulation.