Iterative Domain Decomposition Methods for Singularly Perturbed Nonlinear Convection-Diffusion Equations

We consider special numerical approximations to a domain decomposition method for a boundary value problem in the case of singularly perturbed nonlinear convection-diffusion equations, with the perturbation parameter ε. As a rule, a differential problem is approximated by nonlinear grid equations (iteration-free schemes), which are then solved by suitable iterative methods. In the case of ε-uniform robust methods (c.f. [1]) we require that the solution of the iterative scheme be ε-uniformly close (in the maximum norm) to the solution of the boundary value problem and, furthermore, that the number of iterations required for solving the discrete problem be independent of the parameter ε. Such iterative methods can be effectively solved by means of sequential and parallel domain decomposition algorithms (c.f. [2, 3]). We are interested in constructing schemes of this type which are ε-uniformly robust. In the present paper we construct and study base iterative schemes and their decompositions, i.e. sequential and parallel iterative schemes, for a singularly perturbed nonlinear boundary value problem consisting of an ordinary differential equation of convection-diffusion type. We establish necessary conditions under which the iteration-free scheme is convergent ε-uniformly and the iterative schemes are ε-uniformly robust. The schemes constructed (utilizing piecewise uniform meshes) converge ε-uniformly with the rate O(N−1 lnN) (the convergence rate is optimal or unimprovable (c.f. [4]), the required number of iterations for the iterative schemes is O(lnN), and thus, the ε-uniform amount of computation in solving these iterative schemes is O(N−1 ln N).