SOLVING ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS ON HYPERCUBES.

We discuss the implementation of several classical methods for solving elliptic partial differential equations on multiprocessors based on the hypercube topology. The methods considered are the Alternating Directions Implicit (ADI) algorithm, a direct banded Gaussian elimination method and multigrid methods. The complexity analysis of these algorithms shows that high efficiencies can be achieved by carefully assigning the data to the processors and (sometimes) resorting to more parallelizable methods. The binary reflected Gray code plays an important role for both the multigrid and the ADI algorithms. We shall also briefly discuss implementations of preconditioned conjugate gradient methods and domain decomposition techniques on hypercubes.