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Algebraic Theory of Two-Grid Methods

Published online by Cambridge University Press:  28 May 2015

Yvan Notay
Yvan Notay*
Affiliation:
Université Libre de Bruxelles, Service de Métrologie Nucléaire (C.P. 165-84), 50 Av. F.D.Roosevelt, B-1050 Brussels, Belgium
*
*Email addresses: ynotay@ulb.ac.be (Yvan Notay) Yvan Notay is Research Director of the Fonds de la Recherche Scientifique - FNRS
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Abstract

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.

Keywords

65F0865F1065F50Multigridconvergence analysisalgebraic multigridpreconditioningtwo-level method

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Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 2 , May 2015 , pp. 168 - 198
Copyright
Copyright © Global-Science Press 2015 

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