COLLABORATING PDE SOLVERS WITH INTERFACE RELAXATION

Abstract. This paper deals with one of the well known domain decomposition methods, the collaborating PDE solvers approa.ch. A class of relaxers based on interface relaxation are described. The convergence analysis is pleBen~ed, and the optimal relaxation pa.rameters are deLermined. This analysia applies directly to probleJDs involving Laplace operators, DiricltJ.et boundary conditions, and domains that can be decomposed into re<:ta..tlgles so that each interior corner belongs to ' " rectangles (i.e., interior corners axe ClOSS points). The discrelu:ation used in the analysis is a five-point sta.rl finite difference & on tensor product meshes. The extensions of the analysis to more general problems are discussed, the convergence can be established for more general operators (e.g., self-adjoint or varying from Bubdomain to Iiubdomain), more general boundary conditions (e.g., mixed boundary conditions), more general geometry, and more general discretizations. We are not able to expliciliy solve for optimum relaxation paramelers for these extensions. Numerical results ate also reported. We show that this is a very promising approach to solve complex PDEs on comple:c domains