Parallel Numerical Algorithms Based On Overlapping Schwarz Methods For Singularly Perturbed Semi-Linear Elliptic Equations

. Domain decomposition algorithms based on the Schwarz alternating method are developed for the numerical solution of singularly perturbed semi-linear elliptic problems. The emphasis is given to the description of two variants of the algorithm (multi-domain Schwarz alternating method and two-level Schwarz method) as well as their computer realization on a distributed memory multiprocessor computer. Numerical experiments illustrate the performance of the algorithms on parallel environment as well as their behaviour with respect to the "critical" parameters. Key words. Singularly perturbed elliptic problem, Schwarz domain decomposition methods, parallel computing AMS subject classifications. 35B25, 65N55 1. Introduction. In this paper, we consider the numerical solution of singularly perturbed semi-linear elliptic equations of the form ( Lu(P ) = f(P; u); P = (x; y) 2\Omega ; u(P ) = U(P ); P 2 @\Omega ; (1) posed on a bounded domain\Omega ae R 2 , where L j ¯ 2 (@ 2 =@x ..