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

Abstract

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.

Limit operators, collective compactness, and the spectral theory of infinite matrices

“…finite section and discretisation methods)? We have not tackled this topic in this text, but some results in this direction are in [15,23,70,21] (and see [6,93,79,45,83,84,46,60,85,67,63,19,92,97,98]).…”

“…Anselone and Sloan [6] were the first to extend the arguments of collectively compact operator theory to tackle a case of this type, namely to study the finite section method for classical Wiener-Hopf operators on the half-axis. As mentioned already above, the arguments introduced were developed into a methodology for establishing existence from uniqueness for classes of second kind integral equations on unbounded domains and for analyzing the convergence and stability of approximation methods in a series of papers by the first author and collaborators [15,77,23,27,70,21,26,7,8]. A particular motivation for this was the analysis of integral equation methods for problems of scattering of acoustic, elastic and electromagnetic waves by unbounded surfaces [16,24,112,22,25,70,21,113,75,8,19].…”

“…As mentioned already above, the arguments introduced were developed into a methodology for establishing existence from uniqueness for classes of second kind integral equations on unbounded domains and for analyzing the convergence and stability of approximation methods in a series of papers by the first author and collaborators [15,77,23,27,70,21,26,7,8]. A particular motivation for this was the analysis of integral equation methods for problems of scattering of acoustic, elastic and electromagnetic waves by unbounded surfaces [16,24,112,22,25,70,21,113,75,8,19]. Other applications included the study of multidimensional Wiener-Hopf operators and, related to the Schrödinger operator, a study of Lippmann-Schwinger integral equations [23].…”

Scattering problems from slightly perturbed periodic surfaces: Part II. High order numerical method

“…An alternative way is to apply the variational method, see [CM05,CE10]. The numerical methods for the rough surface scattering problems are always based on the decaying property of the total field (see [MACK00,MC01] for the numerical solutions of the integral equations and [CE10] for the finite section method). Due to the limited decaying rate, the finite section method always converges slowly.…”

Windowed Green Function Method for Layered-Media Scattering

“…The approach proposed in this paper bears similarities with certain "finite-section" methods in the field of rough-surface scattering. These methods utilize approximations based on truncated portions of a given unbounded rough surface [14,22,19] and, in some cases, they incorporate a "taper" [22,21,15] to eliminate artificial reflections from the edges of the finite sections. In fact the smooth taper function utilized in [15] (Figure 2 in that reference) resembles the smooth windowing function we use (Figure 2 below and reference [3]).…”