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

Source abstract

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