On accelerated convergence of nonoverlapping Schwarz methods

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. Schwarz methods for nonoverlapping subdomains also exist but they have not been popular because of their slow convergence. These methods contain a free parameter in the Robin boundary condition of each subdomain problem. The slow convergence can be attributed to an improper choice of this parameter. In this paper, two models are proposed to give guidance to the choice of this parameter. For the Poisson equation on rectangular domains, these models suggest very simple expressions for the parameter in terms of the dimensions of the subdomain. Numerical experiments verify their effectiveness. When used as a preconditioner, it is demonstrated numerically in some examples that the algorithm is quite efficient.