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