On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering

Meier, A. and Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 (2001) On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering. Mathematical Methods in the Applied Sciences, 24 (4). pp. 209-232. ISSN 0170-4214 doi: 10.1002/mma.210

Abstract/Summary

We consider the Dirichlet and Robin boundary value problems for the Helmholtz equation in a non-locally perturbed half-plane, modelling time harmonic acoustic scattering of an incident field by, respectively, sound-soft and impedance infinite rough surfaces.Recently proposed novel boundary integral equation formulations of these problems are discussed. It is usual in practical computations to truncate the infinite rough surface, solving a boundary integral equation on a finite section of the boundary, of length 2A, say. In the case of surfaces of small amplitude and slope we prove the stability and convergence as A→∞ of this approximation procedure. For surfaces of arbitrarily large amplitude and/or surface slope we prove stability and convergence of a modified finite section procedure in which the truncated boundary is ‘flattened’ in finite neighbourhoods of its two endpoints. Copyright © 2001 John Wiley & Sons, Ltd.