Error estimates related to singular perturbations of first-order equations and systems
type: Item
instance of: scholarly article
title: Error estimates related to singular perturbations of first-order equations and systems
zbMATH Open document ID: 0629.35007
DOI: 10.1016/0898-1221(87)90165-9
author: V. Cea
author: C. Bardos
author: Pierre Grisvard
publication date: 1987-01-01T00:00:00Z
review text: Let \(\Omega\) be a bounded open subset in \(R_ 2\), with boundary \(\Gamma\). The homogeneous Dirichlet problem \[ -\epsilon \Delta u_{\epsilon}+a\nabla u_{\epsilon}+bu_{\epsilon}=f\quad in\quad \Omega,\quad u_{\epsilon}=0\quad on\quad \Gamma, \] is studied for \(\epsilon >0\). The solution \(u_{\epsilon}\) is considered as a perturbation of the solution \(u_ 0\) of the mixed problem \(a\nabla u_ 0+bu_ 0=f\) in \(\Omega\), \(u_ 0=0\) on \(\Gamma_-\), where \(\Gamma_- ,\Gamma_+\) and \(\Gamma_ 0\) are the parts of the boundary where \(a\gamma <0\), \(a\gamma >0\) and \(a\gamma =0\), respectively (\(\gamma\) being the outward unit vector field). The authors propose an estimate \[ | u_{\epsilon}(x)-u_ 0(x)| \leq c\{\epsilon +\exp [-\alpha_+(d_+(x)/\epsilon)]+\exp (-\alpha_ 0(d_ 0(x)/\sqrt{\epsilon})]\}, \] where \(d_+\) denotes the distance from x to \(\Gamma_+\), \(d_ 0\) denotes the distance from x to \(\Gamma_ 0\cup \Sigma\) (\(\Sigma\) being the union of all characteristic curves emerging from \(\Gamma_ 0)\). The meaning of this estimate is that \(u_{\epsilon}\) converges toward \(u_ 0\) quickly, far from \(\Gamma_+\cup \Gamma_ 0\cup \Sigma\), but it also converges faster near \(\Gamma_+\) than near \(\Gamma_ 0\cup \Sigma.\) The authors are not able to prove this estimate under general assumptions. They specify, however, rather plausible conditions under which the estimate is valid. The proof is given and discussed in the paper.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 4023806
zbMATH Keywords: error estimates
zbMATH Keywords: homogeneous Dirichlet problem
zbMATH Keywords: perturbation
zbMATH Keywords: mixed problem
MaRDI profile type: Publication
full work available at URL: 0898-1221(87)90165-9
OpenAlex ID: W2071728812
cites work: Regular degeneration and boundary layer for linear differential equations with small parameter
cites work: Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type
cites work: Q5550019
cites work: Q4768140
cites work: Perturbations singulières dans les problèmes aux limites et en contrôle optimal
cites work: Q4098099
cites work: Q4072180
cites work: Q3952494
published in: Computers & Mathematics with Applications
Recommended article: Estimate of the error of an approximate solution of certain singular ordinary differential equations
Recommended article: Q3771517
Recommended article: Discretization of singular systems and error estimation
Recommended article: Error estimates for approximate solutions of equations of the first kind.
Recommended article: Error bound analysis and singularly perturbed Abel-Volterra equations
Recommended article: Error estimates for the finite-element solution of an elliptic singularly perturbed problem
Recommended article: Error estimates for an approximation to a characteristic singular integral equation
Recommended article: On error estimates for approximate solutions of integral equations of the first kind
Recommended article: Error bound for non-zero initial condition using the singular perturbation approximation method
Recommended article: Q3484321
description: scientific article; zbMATH DE number 4023806