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 ^{-1}*N*),*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.