NORMALIZED CONVERGENCE-RATES FOR THE PSMG METHOD

In a previous paper ["Parallel Superconvergent Multigrid," in Multigrid Methods, Marcel Dekker, New York, 1988] the authors introduced an efficient multiscale PDE solver for massively parallel architectures, which was called Parallel Superconvergent Multigrid, or PSMG. In this paper, sharp estimates are derived for the normalized work involved in PSMG solution-the number of parallel arithmetic and communication operations required per digit of error reduction. PSMG is shown to provide fourth-order accurate solutions of Poisson-type equations at convergence rates of .00165 per single relaxation iteration, and with parallel operation counts per grid level of 5.75 communications and 8.62 computations for each digit of error reduction. The authors show that PSMG requires less than one-half as many arithmetic and one-fifth as many communication operations, per digit of error reduction, as a parallel standard multigrid algorithm (RBTRB) presented recently by Decker [SIAM J. Sci. Statist. Comput., 12 (1991), pp. 208-220].