Vorkonditionierung des parallelen Schwarz-Verfahrens zur Lösung parabolischer Differentialgleichungen

The parallel Schwarz method is an important algorithm for the numerical solution of partial differential equations on parallel computers. This method allows on the individual nodes of the parallel computer the usage of well-established sequential software and provides a general framework that can be implemented easily. The fundamental idea leading to an acceleration of the convergence is to proceed in a block Jacobi-like manner at the endpoints of the overlapping intervals. As will be illustrated, the resulting algorithm can be implemented equally easily. The acceleration of the convergence is proven for the one-dimensional heat equation and it is illustrated that for small overlap the convergence rate exceeds that of the classical Schwarz alternating method. Moreover, the equivalence of the derived algorithm and the generalized Schwarz method is shown, leading to a better understanding and handling of the latter method