Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. Abstract M ultigrid methods have been established as being among the most efficient techniques for solving complex elliptic equations. We sketch the multigrid idea emphasizing that multigrid solution is generally obtainable in time directly proportional to the number of unknown variables on serial computers. Despite this, even the most powerful serial computers are not adequate for solving the very large systems generated, for instance, by discretization of fluid flow in three dimensions.A breakthrough can be achieved here only by highly parallel supercomputers. On the other hand, parallel computers are having a profound impact on computational science. Recently, highly parallel machines have taken the lead as the fastest supercomputers, a trend that is likely to accelerate in the future. We describe some of these new computers, and issues involved in using them.We describe standard parallel multigrid algorithms and discuss the question of how to implement them efficiently on parallel machines. The natural approach is to use grid partitioning.One intrinsic feature of a parallel machine is the need to perform interprocessor communication. It is important to ensure that time spent on such communication is maintained at a small fraction of computation time. We analyze standard parallel multigrid algorithms in two and three dimensions from this point of view, indicating that high performance efficiencies are attainable under suitable conditions on moderately parallel machines.We also dernonstrate that such performance is not attainable for multigrid on massively parallel computers, as indicated by an example of poor efficiency on 65,536 processors. The fundamental difficulty is the inability to keep 65,536 processors busy when operating on very coarse grids. This example indicates that the straightforward parallelization of multigrid (and other) algorithms may not always be optimal.However, parallel machines open the possibility of finding really new approaches to solving standard problems. In particular, we present an intrinsically parallel variant of standard multigrid. This "PSMG" method (parallel superconvergent multigrid) allows all processors to be used at all times, even when processing on the coarsest...
