An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods

A domain decomposition method for second-order elliptic problems is considered. An iterative procedure that reduces the problem to a sequence of mixed boundary value problems on each subdomain is proposed. At each iteration, a relaxation is accomplished at the subdomain interfaces. In several circumstances, a value of the relaxation parameter that yields exact convergence in a finite number of iterations is explicitly found. Moreover, when such a value is not available, an appropriate strategy for the automatic selection of the relaxation parameter at each iteration is indicated and analyzed. This iterative method is then applied to the spectral collocation approximation of the differential problem. The same kind of convergence results are proven. Many numerical experiments show the effectiveness of the method proposed here.