Fast adaptive composite grid methods on distributed parallel achitectures

Abstract

AbstractThe fast adaptive composite grid method (FAC) is an algorithm that uses uniform grids, both global and local, to solve partial differential equations. This method is known to be highly efficient on scalar or singe‐processor vector computers, due to its effective use of uniform grids and multiple levels of resolution of the solution. On distributed memory multiprocesssors, such methods benefit from their tendency to create multiple isolated refinement regions, which may be effectively treated in parallel. However, they suffer from the way in which the levels of refinement are treated sequentially in each region. Specifically, the finer levels must wait to be processed until the coarse‐level approximations have been computed and passed to them; conversely, the coarser levels must wait until the finer level approximations have been computed and used to correct their equations.The asynchronous fast adaptive composite method (AFAC) eliminates this bottleneck of parallelism. Through a simple mechanism used to reduce interlevel dependence, individual refinements levels can be processed by AFAC in parallel. The result is that the convergence rate of AFAC is the square root of that for FAC. Therefore, since both AFAc and FAC have roughly the same number of floating‐point operations, AFAC requires twice the serial computational time as FAC, but AFAC, may be much more efficiently parallelized.This paper presents a comparison of FAC and AFAC under a variety of conditions, including vectorization and parallelization. Results are presented for distributed memory multiprocessor architectures (SUPRENUM, Intel iPSC/2 and iPSC/860). The most crucial factor for the evaluation of both algorithms is the different computation/communication cost ratios of these architectures. It is shown that, under many circumstances, AFAC is superior to FAC in a parallel environment.