A generalized Schwarz splitting method based on Hermite collocation for elliptic boundary value problems

Abstract

The Schwarz alternating method (SAM) coupled with various numerical discretization schemes has already been established as an efficient alternative for solving differential equations on various parallel machines. In this paper we consider an extension of SAM (generalized Schwarz splitting—GSS) for solving elliptic boundary value problems with generalized interface conditions that depend on a parameter that might differ in each overlapping region (Tang, 1992). The GSS considered in this paper is coupled with the cubic Hermite collocation discretization scheme (Mitchell et al., 1985) to solve the corresponding boundary value problem in each subdomain. The main objective of this study is the mathematical analysis of the iterative solution of the so-called enhanced GSS collocation discrete matrix equation corresponding to a model elliptic boundary value problem defined on a rectangle. This analysis is based on the spectral properties of the associated enhanced block Jacobi iteration matrix which are explicitly derived. We were able to determine analytically the domain of convergence of the one-parameter GSS scheme for both one-dimensional and two-dimensional problems. In addition sets of optimal multi-parameter GSS schemes have been determined in the case of one-dimensional problems. The analyzed GSS scheme is applied to a number of model elliptic boundary value problems to verify the theoretical results and compare the convergence rates of the SAM and GSS schemes with minimum and maximum overlap. Finally, the same GSS scheme was applied to general elliptic boundary value problems utilizing the optimal interface parameters derived for a model problem. The numerical data obtained indicate that the computational behavior of the optimal GSS schemes determined holds for general elliptic operators.