The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter
type: Item
instance of: scholarly article
title: The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter
zbMATH Open document ID: 0573.65075
DOI: 10.1007/BF01875892
author: Peng-cheng Lin
author: Fawang Liu
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 1984-01-01T00:00:00Z
review text: A homogeneous Dirichlet problem for the sigularly perturbed equation \[ \partial^ 2u/\partial x^ 2+\epsilon \partial^ 2u/\partial y^ 2- a(x,y)\partial u/\partial y=f(x,y),\quad \epsilon >0,\quad a(x,y)>0 \] is considered. When \(\epsilon =0\) this problem reduces to the corresponding boundary value problem for a parabolic equation. The authors seek for a finite difference scheme with the properties: (i) it adapts to the behavior of the boundary layer; (ii) when \(\epsilon\) \(\to 0\) the approximate solution is consistent with the exact solution; (iii) when the step size of the mesh h tends to zero the approximate solution converges uniformly in \(\epsilon\) to the exact solution. The authors construct a difference scheme and find a necessary condition for (iii). Under the assumption that a(x,y) is a constant and f(x,y) is smooth enough this condition is proved to be sufficient for (iii) and the rate of convergence is O(h).
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35M99
zbMATH DE Number: 3915562
zbMATH Keywords: small parameter
zbMATH Keywords: singularly perturbed problem
zbMATH Keywords: uniform convergence
zbMATH Keywords: homogeneous Dirichlet problem
Wikidata QID: Q115393123
reviewed by: Raytcho D. Lazarov
MaRDI profile type: Publication
cites work: A difference scheme for an ordinary differential equation with a small parameter
cites work: Q3909948
cites work: The difference method for the solution of singular perturbation problems for elliptic-parabolic differential equations
cites work: Difference scheme for solving an elliptic equation with small parameter in a domain with curvilinear boundary
full work available at URL: https://doi.org/10.1007/bf01875892
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description: scientific article; zbMATH DE number 3915562