An optimal order collocation method for first kind boundary integral equations on polygons
type: Item
instance of: scholarly article
title: An optimal order collocation method for first kind boundary integral equations on polygons
zbMATH Open document ID: 0827.65117
published in: Numerische Mathematik
publication date: 1995-12-11T00:00:00Z
review text: One of the simplest first kind linear boundary integral equations with a logarithmic kernel, arising in two-dimensional potential theory, is analyzed. It has attracted the intention of many numerical analysts, but despite this a proper explanation of the convergence of the collocation method is still to be given. The authors discuss the convergence of the collocation method using splines of any order \(k\) for such integral equations on closed polygonal boundaries in \(\mathbb{R}^2\). Before discretization the equation is transformed to an equivalent equation over \([-\pi, \pi]\) using a nonlinear parametrization of the polygon. This has the effect of producing a transformed equation with a solution which is smooth on \([- \pi,\pi]\). The structure of the integral operator is described in detail. The authors show that the collocation method using splines of order \(k\) converges with optimal order \(O(h^k)\). Using the numerical solutions of the transformed equation, numerical solutions of the original equation are constructed which converge optimally in a certain weighted norm. The convergence results are illustrated with some numerical examples.
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 31A30
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 31A10
zbMATH DE Number: 739394
zbMATH Keywords: first kind linear boundary integral equation
zbMATH Keywords: logarithmic kernel
zbMATH Keywords: potential theory
zbMATH Keywords: convergence
zbMATH Keywords: collocation method
zbMATH Keywords: numerical examples
author: Johannes Elschner
author: I. G. Graham
reviewed by: T. Burczynski
MaRDI profile type: Publication
full work available at URL: s002110050107
OpenAlex ID: W2015466529
DOI: 10.1007/S002110050107
description: scientific article; zbMATH DE number 739394