Multi-parameterized Schwarz alternating methods for elliptic boundary value problems

The convergence rate of a numerical procedure based on Schwarz Alternating 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) = ωu + (1 − ω)ϖuϖn), controlled by the parameter ω, can optimize SAMs convergence rate. In this paper, we present a matrix formulation of this method based on finite difference approximation of the BVP, review its known computational behavior in terms of the parameter α = /gf(ω, h), where h is the discretization parameter and /gf is a derivable relation, and obtain analytically explicit and implicit expressions for the optimum α. Moreover, we consider a parameterized SAM where the parameter ω or α is assumed to be different in each overlapping area. For this SAM and the one-dimensional (1-D) elliptic model BVPs, we determine analytically the optimal values of αi. Furthermore, we extend some of these results to two-dimensional (2-D) elliptic problems.