NOTE ON THE PARALLEL EFFICIENCY OF THE FREDERICKSON-MCBRYAN MULTIGRID ALGORITHM

Standard multigrid algorithms must lead to processor idle time on large-scale parallel computers because the coarsest grids have fewer points than processors. In some cases, this may be considered to be a disadvantage. Frederickson and McBryan [Multigrid Methods, Marcel Dekker, New York, 1988] show that retaining all points on all grid levels (using all processors) can lead to a "superconvergent" algorithm in that a very good convergence rate is obtained. Has the "parallel superconvergent" multigrid algorithm (PSMG) of Frederickson and McBryan solved the problem of implementing multigrid on a massively parallel single-instruction-multiple-data (SIMD) architecture? How much can be gained by retaining all points on all grid levels, keeping all processors busy? The purpose of this note is to compare the parallel efficiency of the PSMG algorithm to a standard multigrid algorithm. It is shown that the perfect processor utilization and the good convergence rates of the PSMG algorithm do lead to a more efficient algorithm for the special case of one (or fewer) grid points per processor. Normalized computation and communication requirements are given, so that the two types of algorithms can be compared directly.