Multi-Parameterized Schwarz Splittings

Numerical versions of the classical mathematical approach of Schwarz Alternating Method (SAM), introduced by Schwarz (15] a century ago, have been recently explored as parallel computational frameworks for the numerical solut.ion of initial/boundary value problems. These schemes are usually referred to as Schwarz Splitlings(.'J'S). One of the uncertainties in the numerical formulation of SAM that affects its convergence is the selection of the so called interface conditions. In the context of elliptic boundary value problems the most commonly used auxiliary conditions are of Dirichlet t.ype. In this paper we consider one-parameter (a) mixed interface conditions (i.e. the same parameter is used in all sub domains) and the multi -parameter case where a different parameter (ni) is associated with the i-th overlapping area. The SS approach with the mixed boundary conditions is referred to as Generalized(G) SS. An appropriate choice of the parameter n relating the weights between the Dirichlet and the Neumann conditions allows one to optimize the convergence rates of the GSS [17]. In this paper, first we determine explicitly the optimal value of n for the cases of two and three subregions and two-point boundary value problems and propose an approach for determining a experimentally in the case of more than three subregions. Second, we successfully determine the optimal values of a;'s for which the spectral radius of t.he block Jacobi iteration matrix associated with the GSS is zero. The parametrized .'is methods studied in this paper are extended and studied for twoand higher-dimensional boundary value problems in [7].