Residual Scaling Techniques in Multigrid, II: Practical Applications

Jun Zhang
Department of Mathematics
The George Washington University
Washington, DC, USA
Department of Computer Science and Engineering
University of Minnesota
Minneapolis, MN 55455, USA


This paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of an earlier companion paper: Residual Scaling Techniques in Multigrid, I: Equivalence Proof. We discuss the computational issues of some residual scaling techniques which have been proved mathematically equivalent in the companion paper. A heuristic residual analysis technique, based on the geometry of the grid points and the relaxation pattern, is introduced to estimate the optimal residual scaling factor for a high-order multigrid method. We compare the performance of a typical pre-optimization (pre-acceleration) technique with a typical post-optimization (post-acceleration) technique and show that the pre-optimization is preferable in both convergence and efficiency. Our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre-optimization technique over the post-optimization technique.

Key words and phrases: Multigrid method, minimal residual smoothing, residual scaling techniques, heuristic residual analysis.

Contributed September 28, 1997.