When does two-grid optimality carry over to the V-cycle?

Abstract

We investigate additional condition(s) that confirm that a V-cycle multigrid method is satisfactory (say, optimal) when it is based on a two-grid cycle with satisfactory (say, level-independent) convergence properties. The main tool is McCormick's bound on the convergence factor (SIAM J. Numer. Anal. 1985; 22:634-643), which we showed in previous work to be the best bound for V-cycle multigrid among those that are characterized by a constant that is the maximum (or minimum) over all levels of an expression involving only two consecutive levels; that is, that can be assessed considering only two levels at a time. We show that, given a satisfactorily converging two-grid method, McCormick's bound allows us to prove satisfactory convergence for the V-cycle if and only if the norm of a given projector is bounded at each level. Moreover, this projector norm is simple to estimate within the framework of Fourier analysis, making it easy to supplement a standard two-grid analysis with an assessment of the V-cycle potentialities. The theory is illustrated with a few examples that also show that the provided bounds may give a satisfactory sharp prediction of the actual multigrid convergence. © 2010 John Wiley & Sons, Ltd.