Domain Decomposition Method for a Semilinear Singularly Perturbed Elliptic Convection-Diffusion Equation with Concentrated Sources

In the case of regular boundary value problems, effective numerical methods based on domain decomposition are well known (see, for example, [1, 2] and the list of references there). Such methods allows us to reduce the solution of complicated problems with several characteristic scales to solving simplified problems on simpler subdomains and, in particular, to implement parallel computations. Errors of standard numerical methods applied to singularly perturbed problems can many times exceed the solution itself for small values of the singular perturbation parameter ε. For this type of problems, it is of interest to develop numerical methods whose errors are independent of the parameter ε and defined only by the number of mesh points (that is, such methods are said to converge ε-uniformly). When using an iterative method for solving nonlinear problems and/or domain decomposition schemes, we will require that its accuracy is defined only by the number of mesh points, moreover, the number of iterations required for convergence is also independent of ε. In the case of nonlinear singularly perturbed problems, the attainment of ε-uniform convergence of a numerical methods necessarily requires the use of special meshes whose step-size in a boundary (interior) layer is exceedingly small (see, e.g., [3]). When such problems are decomposed, the number of iterations in the corresponding iterative process can be large and essentially depend on the parameter ε. Therefore, it is important to develop numerical methods based on domain decomposition techniques that converge ε-uniformly. For singularly perturbed problems in a composed domain (in particular, with concentrated sources) whose solution has several singularities such as boundary and interior layers, it is of keen interest to construct such methods so that each subdomain in the domain decomposition contains no more than a single singularity, which essentially simplifies the solution of the problem under consideration. In this paper, we develop monotone linearized schemes based on an overlapping Schwarz method for a semilinear singularly perturbed elliptic convection-diffusion equation in a composed domain (vertical strip) in the presence of a concentrated source acting on the interface. We first study a special (base) scheme comprising a standard finite difference operator on a piecewiseuniform mesh condensing in the boundary and interior layers, and then an overlapping domain decomposition scheme constructed on the basis of the former that converge ε-uniformly at the rates O ( N 1 lnN1 +N −1 2 )