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On Conditioning of a Schwarz Method for Singularly Perturbed Convection–diffusion Equations in the Case of Disturbances in the Data of the Boundary-value Problem by Gregorii I. Shishkin is a Engineering article available to read on EtoBox.

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## Abstract In this paper we discuss conditioning of a discrete Schwarz method on piecewise–uniform meshes with an example of a one-dimensional singularly perturbed boundary-value problem. We consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter ε. To solve the problem numerically we use an ε-uniformly convergent (in the maximum norm) difference scheme on special piecewise–uniform meshes. For this base scheme we construct a decomposition scheme based on a Schwarz technique with overlapping subdomains, which converges ε-uniformly with respect to both the number of mesh points and the number of iterations. The step-size of such special meshes is extremely small in the neighborhood of the layer and changes sharply on its boundary, that (as was shown by A.A. Samarskii) can generally lead to a loss of well-conditioning of the above schemes. For the decomposition scheme we study the conditioning of the system (difference scheme) and the conditioning of the system matrix (difference operator), and also the influence of perturbations in the data of the boundary-value problem on disturbances of its nume

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Author

Gregorii I. Shishkin

Publisher

Walter de Gruyter GmbH

Published

2003

Language

EN

Field

Engineering (Physical Sciences)