A finite element method for a singularly perturbed boundary value problem
type: Item
instance of: scholarly article
title: A finite element method for a singularly perturbed boundary value problem
zbMATH Open document ID: 0583.65054
DOI: 10.1007/BF01389664
author: Martin Stynes
author: Eugene O'Riordan
published in: Numerische Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133135
review text: We examine the problem: \(\epsilon u''+a(x)u'-b(x)u=f(x)\) for \(0<x<1\), \(a(x)\geq \alpha >0\), b(x)\(\geq \beta\), \(\alpha^ 2+4\epsilon \beta >0\), a, b and f in \(C^ 2[0,1]\), \(\epsilon\) in (0,1], u(0) and u(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of width h is uniformly second order accurate for this problem (i.e., the nodal errors are bounded by \(Ch^ 2\), where C is independent of h and \(\epsilon)\). With a natural choice of trial functions, uniform first order accuracy is obtained in the \(L^{\infty}(0,1)\) norm. On choosing piecewise linear trial functions (''hat'' functions), uniform first order accuracy is obtained in the \(L^ 1(0,1)\) norm.
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 3934339
zbMATH Keywords: hat function
zbMATH Keywords: finite elements
zbMATH Keywords: difference scheme
zbMATH Keywords: uniform first order accuracy
zbMATH Keywords: ''hat'' functions
MaRDI profile type: Publication
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OpenAlex ID: W2006242933
description: scientific article; zbMATH DE number 3934339