Report 2009-27

Approximation by plane waves

A. Moiola, R. Hiptmair and I. Perugia

Abstract: This paper studies the approximation of the solutions of the homogeneous Helmholtz equation by finite dimensional spaces of plane, circular and spherical wave functions. The main results are the proofs of:

- algebraic convergence in the domain size h in two and three dimensions;
- algebraic convergence in the number p of approximating functions in two dimensions.

The approximation error is measured in weighted Sobolev norms; the dependence of all bounds on the wavenumber is made explicit. The proofs rely on an explicit formulation of Vekua's theory for Ndimensional Helmholtz equation (N \ge 2) and on approximation properties for harmonic functions. The obtained estimates can be used in the analysis of the convergence of several Trefftz-type finite elements methods.

Paper: Available as PDF (3412KB) or as hardcopy to order reports@sam.math.ethz.ch.