Iterative grid methods for singularly perturbed elliptic equations degenerating into zero-order ones

Iterative grid methods for singularly perturbed elliptic equations degenerating into zero-order ones - A Dirichlet problem is considered on a rectangle for singularly perturbed linear and quasi-linear elliptic equations. When the perturbation parameter equals zero, elliptic equations degenerate into zeroorder ones. Special iteration-free and iterative difference schemes which converge uniformly with respect to the parameter are constructed for boundary-value problems. Schwarz* method and the domain decomposition method are used to construct the schemes. Necessary and sufficient conditions are given for the solutions of the iterative difference schemes to converge uniformly with respect to the perturbation parameter as the number of iterates increases. Various approaches to forming difference equations have been developed in order to construct grid approximations to problems of mathematical physics, in particular, to the equations that have smooth enough coefficients and solutions (see, for example, [9,10]). Methods of constructing difference schemes on the basis of alternating Schwarz' method and the domain decomposition method are fairly attractive, with them we can reduce the original problem to a sequence of subproblems on subdomains of a simpler form [5,19]. In addition, these methods allow parallel computation on multiprocessor computers [9]. The solutions of singularly perturbed boundary-value problems are of limited smoothness, and the accuracy of an approximate solution obtained with a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Russian Journal of Numerical Analysis and Mathematical Modelling de Gruyter

Iterative grid methods for singularly perturbed elliptic equations degenerating into zero-order ones

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0927-6467
eISSN
1569-3988
DOI
10.1515/rnam.1993.8.4.341
Publisher site
See Article on Publisher Site

Abstract

- A Dirichlet problem is considered on a rectangle for singularly perturbed linear and quasi-linear elliptic equations. When the perturbation parameter equals zero, elliptic equations degenerate into zeroorder ones. Special iteration-free and iterative difference schemes which converge uniformly with respect to the parameter are constructed for boundary-value problems. Schwarz* method and the domain decomposition method are used to construct the schemes. Necessary and sufficient conditions are given for the solutions of the iterative difference schemes to converge uniformly with respect to the perturbation parameter as the number of iterates increases. Various approaches to forming difference equations have been developed in order to construct grid approximations to problems of mathematical physics, in particular, to the equations that have smooth enough coefficients and solutions (see, for example, [9,10]). Methods of constructing difference schemes on the basis of alternating Schwarz' method and the domain decomposition method are fairly attractive, with them we can reduce the original problem to a sequence of subproblems on subdomains of a simpler form [5,19]. In addition, these methods allow parallel computation on multiprocessor computers [9]. The solutions of singularly perturbed boundary-value problems are of limited smoothness, and the accuracy of an approximate solution obtained with a

Journal

Russian Journal of Numerical Analysis and Mathematical Modellingde Gruyter

Published: Jan 1, 1993

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