Wavenumber Explicit Convergence Analysis for Galerkin Discretizations of the Helmholtz Equation

In this paper, we develop a new stability and convergence theory for highly indefinite elliptic partial differential equations by considering the Helmholtz equation at high wave number as our model problem. The key element in this theory is a novel k-explicit regularity theory for Helmholtz boundary value problems that is based on decomposing the solution into in two parts: the first part has the H2-Sobolev regularity expected of elliptic PDEs but features k-independent regularity constants; the second part is an analytic function for which k-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of boundary value problems including the case Robin boundary conditions in domains with analytic boundary and in convex polygons. As the most important practical application we apply our full error analysis to the classical hp-version of the finite element method (hp-FEM) where the dependence on the mesh width h, the approximation order p, and the wave number k is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems grows only polynomially in k, 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). AMS Subject Classification: 35J05, 65N12, 65N30