A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane
type: Item
instance of: scholarly article
title: A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane
zbMATH Open document ID: 1012.65145
author: Yanyan Li
published in: Numerische Mathematik
publication date: 2002-12-01T00:00:00Z
review text: The numerical treatment of second kind integral equations on the real line \[ \varphi(s)=\psi(s)+\int_{-\infty}^{+\infty}k(s-t)z(t)\varphi(t) dt,\quad s\in \mathbb{R}, \] is considered where \(k\in L_1(\mathbb{R}),z\in L_{\infty}(\mathbb{R})\)\ and \(\psi\in BC(\mathbb{R})\) (bounded continuous functions on \(\mathbb{R}\)). \(\varphi\) is the unknown function. A sharp error estimate for the finite section approximation is derived. The numerical solution by a simple discrete collocation method on a uniform grid on \(\mathbb{R}\) is analysed. The case when \(z\)\ is compactly supported leads to a coefficient matrix allowing a rapid matrix vector multiplication by the fast Fourier transform and a modified two-grid iteration is used. If \(z\)\ is not compactly supported a finite section and a two-grid algorithm are used. As an application an acoustic scattering problem is presented. An error estimate of the numerical solution is given.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 76Q05
zbMATH DE Number: 1837415
zbMATH Keywords: fast two-grid method
zbMATH Keywords: finite section method
zbMATH Keywords: acoustic scattering
zbMATH Keywords: convolution type integral equation of the second kind
zbMATH Keywords: error estimate
zbMATH Keywords: discrete collocation method
zbMATH Keywords: two-grid iteration
zbMATH Keywords: fast Fourier transform
reviewed by: Josef Kofroň
MaRDI profile type: Publication
full work available at URL: s002110100373
OpenAlex ID: W4252716533
DOI: 10.1007/S002110100373
description: scientific article; zbMATH DE number 1837415