The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system
type: Item
instance of: scholarly article
title: The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system
zbMATH Open document ID: 1068.65099
author: Xianqiang Yang
published in: Applied Mathematics and Computation
publication date: 2005-02-23T00:00:00Z
review text: The numerical solution for singularly perturbed initial boundary value problem of the system of nonlinear differential equations of the form \[ \begin{cases} \varepsilon u_1'+f_1(x,u_1,u_2)=0,&0<x\leq\ell\\ -u_2''+f_2(x,u_1,u_2)=0,&0<x<\ell\\ u_1(0)=u_*,\\ u_2(0)=A,\;u_2(\ell)=B\end{cases} \tag{\(*\)} \] is obtained, where \(0<\varepsilon<1\) is a small parameter, \(u_*\), \(A, B\) are given constants, \(f_1\) and \(f_2\) are given smooth functions satisfying certain regularity conditions. A method based on finite elements with piecewise constant and piecewise linear basis functions and appropriate quadrature formulae with remainder term in integral form is derived. It is shown that the solution of the finite difference scheme on a special non-uniform mesh converges point-wise independently of the singular perturbation parameter. Numerical results supporting the theory developed are presented.
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L10
zbMATH DE Number: 2137733
zbMATH Keywords: singular perturbation
zbMATH Keywords: non-uniform mesh
zbMATH Keywords: difference scheme
zbMATH Keywords: uniform convergence
zbMATH Keywords: boundary value problem
zbMATH Keywords: system
zbMATH Keywords: nonlinear
zbMATH Keywords: finite elements
zbMATH Keywords: numerical results
MaRDI profile type: Publication
full work available at URL: j.amc.2004.01.015
OpenAlex ID: W2045633467
cites work: Approximate solution of a non-linear boundary value problem with a small parameter for the highest-order differential
cites work: The solution of singularly perturbed huge problems via domain decomposition
cites work: Q4485828
cites work: Q4871736
DOI: 10.1016/J.AMC.2004.01.015
description: scientific article; zbMATH DE number 2137733