Numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method
type: Item
instance of: scholarly article
title: Numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method
zbMATH Open document ID: 0624.65114
DOI: 10.1016/0307-904X(86)90057-0
author: R. Rangogni
published in: Applied Mathematical Modelling
publication date: 1986-01-01T00:00:00Z
review text: The generalized Laplace equation of the title is \(\nabla.(a\nabla v)=0\) which is to be solved in a bounded region D of \({\mathbb{R}}^ 2\) subject to mixed boundary conditions. It is assumed that \(a(x,y)>0\) in D. Set \(v=u.a^{-}\), then u satisfies an equation in D of the form \(\nabla^ 2u+fu=0.\) The proposal is to solve the equation \(\nabla^ 2\omega +\epsilon f\omega =0\) by expanding its solution as \(\sum^{\infty}_{j=0}\epsilon^ ju_ j.\) This leads to an infinite sequence of coupled boundary value problems for \(u_ 0,u_ 1,...\), each of which will satisfy a Poisson equation. The required solution should be \(\sum^{\infty}_{j=0}u_ j.\) There is a brief discussion of the numerical methods which were used to solve the Poisson equations. The vital problem of the convergence of the solution series is not considered. The author comments however that for the two worked examples for which the solution is known only two terms of the series sufficed. The results are not wholly convincing.
reviewed by: D. Kershaw
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35C15
zbMATH DE Number: 4013951
zbMATH Keywords: perturbation method
zbMATH Keywords: series expansion
zbMATH Keywords: boundary element method
zbMATH Keywords: generalized Laplace equation
zbMATH Keywords: Poisson equation
zbMATH Keywords: convergence
MaRDI profile type: Publication
cites work: Q5821083
cites work: Numerical solution of the generalized Laplace equation by the boundary element method
cites work: Q3049321
cites work: Q3888653
cites work: The solution of the non-homogeneous Helmholtz equation by means of the boundary element method
description: scientific article; zbMATH DE number 4013951