Analysis of Minimal Residual Smoothing in Multigrid

Jun Zhang
Department of Mathematics
The George Washington University
Washington, DC 20052, USA


We analyze the multigrid method accelerated by a minimal residual smoothing (MRS) technique. We prove that the MRS acceleration scheme is a semi-iterative method with respect to the original multigrid method and that the MRS accelerated multigrid method is a polynomial acceleration of first order. We explain the situations that MRS acceleration rate may slow down. The iteration matrices for the MRS accelerated coarse-grid-correction operator and the MRS accelerated two-level operator are obtained. In a simplified model, we give conditions for accelerating two-level method and some estimates for the acceleration rate. These analytical estimates agree quite well with our numerical results reported in an early paper: Minimal Residual Smoothing in Multi-level Iterative Method. The discussions in this paper are theoretical and are focused on the two-level method because MRS is only applied on the finest level of the multigrid method.

Key words: Minimal residual smoothing, multigrid method, residual transfer, conjugate gradient-type methods.

AMS subject classifications: 65F10, 65N06.

Contributed February 26, 1996.