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

In this paper we consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter e. To solve the problem numerically we use an e-uniformly convergent difference scheme (on special piecewise-uniform meshes) and a decomposition of this scheme based on a Schwarz technique with overlapping subdomains. The step-size of such special meshes is extremely small in a neighbourhood of the layer and changes sharply on its boundary, which can generally lead to a loss of conditioning of the above schemes. We study the influence of perturbations in the data of the boundary value problem on disturbances of numerical solutions. We derive estimates for the disturbances of numerical solutions (in the maximum norm) depending on a subdomain in which the disturbance of the data appears. 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 (that is, e ln N, for e = O(ln-1N), in a neighbourhood of the boundary layer, where N defines the number of mesh points), the finite difference schemes under consideration are well conditioned e-uniformly. In addition, for the Schwarz method a special restriction is imposed on the width of the overlapping region. Note that for these special schemes an e-uniform estimate for the condition number is the same as that for schemes on uniform meshes in the case of regular boundary value problems. We give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent e-uniformly to the solution of the Dirichlet problem as the number of mesh points and the number of iterations increase.