caa a22 4500 378857711 CHVBK 20180305123339.0 cr unu---uuuuu 161128s2003 xx s 000 0 eng 10.2478/cmam-2003-0030 doi (NATIONALLICENCE)gruyter-10.2478/cmam-2003-0030 Shishkin Gregorii I. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., 620219 Ekaterinburg, Russia. 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 [Elektronische Daten] [Gregorii I. Shishkin] 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 numerical solutions. We derive estimates for the disturbances of the numerical solutions (in the maximum norm) depending on the subdomain in which the disturbance of the data appears. It is shown that the condition number of the difference operator associated with the Schwarz method, just as for the base scheme, is not ε-uniformly bounded. However, these difference schemes are well-conditioned ε-uniformly (with the ε-uniform estimate for the condition number being the same as for the schemes on uniform meshes for regular problems) when the right-hand side of the discrete equations is considered in a "natural” norm, i.e., in the maximum norm with a special weight multiplier. In the case of the boundary-value problem with perturbed data we give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent ε-uniformly to the solution of this Dirichlet problem as the number of mesh points and the number of iterations increase. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. singular perturbation problem nationallicence convection-diffusion equations nationallicence ε-uniform fitted mesh method nationallicence domain decomposition method nationallicence condition numbers nationallicence Computational Methods in Applied Mathematics De Gruyter 3/3(2003), 459-487 1609-4840 3:3<459 2003 3 cmam https://doi.org/10.2478/cmam-2003-0030 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.2478/cmam-2003-0030 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Shishkin Gregorii I. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., 620219 Ekaterinburg, Russia NATIONALLICENCE 773 0- Computational Methods in Applied Mathematics De Gruyter 3/3(2003), 459-487 1609-4840 3:3<459 2003 3 cmam CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter