The efficient parallel solution of PDEs

The report presents some results in solving finite element equations via a parallel version of the preconditioned cg-method (ParPCG). We use a nonoverlapping domain decomposition and construct preconditioners based on Additive and Multiplicative Schwarz Methods (ASM/MSM). As components in the preconditioner, multigrid methods, hierarchical bases, new extension techniques and modified BPS- and BPX-preconditioners for handling the unknowns at the coupling nodes on the boundaries between subdomains are used. The scale up efficiency (e.g., an increasing number of processors causes an increasing problem size) of the algorithm by doubling the number of processors is larger than 95%. Even the practical not relevant speed up efficiency (e.g., an increasing number of processors and a constant problem size) reaches 80% by doubling the number of processors.