Parallel implicit domain-decomposed solvers for PDEs

Parallel computing is providing the next advance in computer hardware and software for the solution of partial differential equations. The real potential in these advances is in massively parallel computing (the next generation of supercomputers) and in distributed computing (large clusters of workstations). In both of these cases, algorithms that are efficient for these machines must make efficient use of local memory and restrict the use of distant memory, while not sacrificing overall performance. Current research into domain-decomposition algorithms has shown how to efficiently combine a small amount of global information with purely local information to construct efficient solvers for systems of linear equations arising from PDEs. This paper will review the current state of the art and discuss some aspects of software that implements these algorithms.