Two-level parallel Schwarz methods for singularly perturbed semilinear elliptic problems

Two-level domain decomposition methods based on the Schwarz alternating procedure are proposed for the numerical solution of singularly perturbed semilinear elliptic problems. Convergence properties of the two-level Schwarz methods are established. Numerical experiments for a test problem illustrate the behaviour of the methods with respect to the critical parameters, such as the perturbation parameter, the amount of overlapping and sizes of subdomains. Application of one of the two-level Schwarz methods to a model semiconductor problem is also discussed.