Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions

A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in ${\mathbb {R}}^{d}$, $d \in \{1,2,3\}$ is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical $hp$-version of the finite element method ($hp$-FEM) is presented for the model problem where the dependence on the mesh width $h$, the approximation order $p$, and the wave number $k$ is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least $O(\log k)$.