J. Melenk, S. Sauter:

"Wave-number explicit convergence analysis for Galerkin discretizations of the Helmholtz equation (extended version)";

in: "ASC Report 31/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

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).

Helmholtz equation at high wavenumber, stability, convergence, hp−finite elements

http://www.asc.tuwien.ac.at/preprint/2009/asc31x2009.pdf

Created from the Publication Database of the Vienna University of Technology.