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When does two‐grid optimality carry over to the V‐cycle?
Author(s) -
Napov Artem,
Notay Yvan
Publication year - 2010
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.685
Subject(s) - multigrid method , mathematics , convergence (economics) , projector , norm (philosophy) , upper and lower bounds , bounded function , grid , simple (philosophy) , uniform convergence , mathematical analysis , computer science , geometry , partial differential equation , computer network , philosophy , epistemology , bandwidth (computing) , political science , law , economics , computer vision , economic growth
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. Copyright © 2009 John Wiley & Sons, Ltd.