A GENERALIZED SCHWARZ SPLITTING METHOD BASED ON HERMITE COLLOCATION FOR ELLIPTIC BOUNDARY-VALUE-PROBLEMS

The Schwarz alternating method (SAM) coupled with various numerical di scretization schemes has already been established as an efficient alte rnative for solving differential equations on various parallel machine s. In this paper we consider an extension of SAM (generalized Schwarz splitting-GSS) for solving elliptic boundary value problems with gener alized interface conditions that depend on a parameter that might diff er in each overlapping region (Tang, 1992). The GSS considered in this paper is coupled with the cubic Hermite collocation discretization sc heme (Mitchell et al., 1985) to solve the corresponding boundary value problem in each subdomain. The main objective of this study is the ma thematical analysis of the iterative solution of the so-called enhance d GSS collocation discrete matrix equation corresponding to a model el liptic boundary value problem defined on a rectangle. This analysis is based on the spectral properties of the associated enhanced block Jac obi iteration matrix which are explicitly derived. We were able to det ermine analytically the domain of convergence of the one-parameter GSS scheme for both one-dimensional and two-dimensional problems. In addi tion sets of optimal multi-parameter GSS schemes have been determined in the case of one-dimensional problems. The analyzed GSS scheme is ap plied to a number of model elliptic boundary value problems to verify the theoretical results and compare the convergence rates of the SAM a nd GSS schemes with minimum and maximum overlap. Finally, the same CSS scheme was applied to general elliptic boundary value problems utiliz ing the optimal interface parameters derived for a model problem. The numerical data obtained indicate that the computational behavior of th e optimal GSS schemes determined holds for general elliptic operators.