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On the stability and convergence of the finite section method for integral equation formulations of rough surface scattering
Author(s) -
Meier A.,
ChandlerWilde S. N.
Publication year - 2001
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.210
Subject(s) - mathematics , helmholtz equation , mathematical analysis , integral equation , boundary value problem , electric field integral equation , boundary (topology) , stability (learning theory) , dirichlet boundary condition , machine learning , computer science
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 2 A , 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.