MULTI-PARAMETERIZED SCHWARZ ALTERNATING METHODS FOR ELLIPTIC BOUNDARY-VALUE-PROBLEMS

The convergence rate of a numerical procedure based on Schwarz Alterna ting Method (SAM) for solving elliptic boundary value problems (BVPs) depends on the selection of the so-called interface conditions applied on the interior boundaries of the overlapping subdomains. It has been observed that the weighted mixed interface conditions (g(u) = omega u + (1 - omega)partial derivative u/partial derivative n), controlled b y the parameter omega, can optimize SAMs convergence rate. In this pap er, we present a matrix formulation of this method based on finite dif ference approximation of the BVP, review its known computational behav ior in terms of the parameter alpha = phi(omega, h), where h is the di scretization parameter and phi is a derivable relation, and obtain ana lytically explicit and implicit expressions for the optimum alpha. Mor eover, we consider a parameterized SAM where the parameter omega or al pha is assumed to be different in each overlapping area. For this SAM and the one-dimensional (1-D) elliptic model BVPs, we determine analyt ically the optimal values of alpha(i). Furthermore, we extend some of these results to two-dimensional (2-D) elliptic problems.