Parallel algorithms for elliptic equations

Abstract

AbstractWe describe a number of parallel algorithms for the numerical solution of elliptic equations and present some implementations on the Denelcor HEP Parallel Processor.The equations we study, typical of those encountered in a range of fluid pressure equations, are of the form magnified image where the coefficient tensor may represent a density, permeability or dielectric constant. We allow the possibility of discontinuities in , and of line and point sources in F. The equations are discretized using finite elements, with orders ranging from linear to bicubic.We use a tree of increasingly fine grids to achieve parallelism in the course of mesh refinement. We use conjugate gradient methods as well as multigrid methods to solve the resulting algebraic equations within each subgrid. We describe the implementation of these methods as parallel algorithms, and present results of actual performance on the HEP computer. In each case we demonstrate near optimal performance on the HEP, as measured by speedup over the corresponding serial algorithm.We would like to thank both Argonne National Laboratory and Ballistic Research Laboratory for allowing us access to their HEP computers. The operations staffs at both sites gave us unstintingly of their time, even when called on after hours. This research, on an, at times, quite temperamental machine, would never have been completed without their help. We would also like to thank Argonne National Laboratory for the excellent two‐day HEP introduction provided us prior to the start of this research. Finally, we wish to thank E. Lusk and R. Overbeek for providing us with copies of their HEP macros.