CONVERGENCE ANALYSIS FOR FINITE ELEMENT DISCRETIZATIONS OF THE HELMHOLTZ EQUATION WITH DIRICHLET-TO-NEUMANN BOUNDARY CONDITIONS

摘要

A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R~d, d ∈ {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).