Strong uniform stability and exact discretizations of a model singular perturbation problem and its finite-difference approximations
type: Item
instance of: scholarly article
title: Strong uniform stability and exact discretizations of a model singular perturbation problem and its finite-difference approximations
zbMATH Open document ID: 0681.65059
DOI: 10.1016/0096-3003(89)90135-5
published in: Applied Mathematics and Computation
publication date: 1989-01-01T00:00:00Z
review text: The author studies both the strong uniform stability and the norm \(\|| u\||_{\epsilon}:=\| u\|_{\infty}+\delta \| u'\|_{\infty}\) and three-point approximation in the norm \(\|| u^ H\||_{H,\epsilon}:=\| u^ H\|_{H,\infty}+\epsilon \| Du^ H\|_{H,\infty}\) for the model problem \[ (1)\quad L_{\epsilon}u:=-\epsilon u''+a(x)u'+b(x)u=f(x),\quad 0<x<1,\quad u(0)=A,\quad u(1)=B, \] where \(\epsilon >0\), \(a\in L^{\infty}[0,1]\), \(b,f\in L^ 1[0,1]\), \(a(x)\geq a>0\). The strong uniform stability of the discrete schemes is analyzed using exact three-point schemes for (1). It should be pointed out that the exact three-point difference schemes for differential equations of second order were introduced by \textit{A. N. Tikhonov} and \textit{A. A. Samarskii} [Zh. Vychisl. Mat. Mat. Fiz. 1, 425- 440 (1961; Zbl 0125.073)]. Three-point difference approximation of (1), which have been build on this ground and possess uniform speed of convergence were given by \textit{M. V. Alekseevskij} [Difference methods in mathematical physics, Collect. Pap., Moscow 1979, 36-60 (1979; Zbl 0456.34022)].
reviewed by: Vladimir L. Makarov
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 4115868
zbMATH Keywords: singular perturbation
zbMATH Keywords: exact discretization
zbMATH Keywords: strong uniform stability
zbMATH Keywords: exact three-point schemes
author: E. C. jun. Gartland
MaRDI profile type: Publication
full work available at URL: 0096-3003(89)90135-5
OpenAlex ID: W4247293256
cites work: An Analysis of a Uniformly Accurate Difference Method for a Singular Perturbation Problem
cites work: Discrete Weighted Mean Approximation of a Model Convection-Diffusion Equation
cites work: Uniform High-Order Difference Schemes for a Singularly Perturbed Two-Point Boundary Value Problem
cites work: An Analysis of a Uniformly Convergent Finite Difference/Finite Element Scheme for a Model Singular-Perturbation Problem
cites work: Graded-Mesh Difference Schemes for Singularly Perturbed Two-Point Boundary Value Problems
cites work: Strong Stability of Compact Discrete Boundary Value Problems via Exact Discretizations
cites work: Analysis of Some Difference Approximations for a Singular Perturbation Problem Without Turning Points
cites work: Die gleichmäßige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme
cites work: An Analysis of a Superconvergence Result for a Singularly Perturbed Boundary Value Problem
cites work: Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems
description: scientific article; zbMATH DE number 4115868