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Algebraic analysis of two-grid methods: the nonsymmetric case

par Notay, Yvan
Référence Numerical linear algebra with applications, 17, page (73-96)
Publication Publié, 2010

Article révisé par les pairs

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Résumé :

Two-grid methods constitute the building blocks of multigrid methods, which are among the most efficient solution techniques for solving large sparse systems of linear equations. In this paper, an analysis is developed that does not require any symmetry property. Several equivalent expressions are provided that characterize all eigenvalues of the iteration matrix. In the symmetric positive-definite (SPD) case, these expressions reproduce the sharp two-grid convergence estimate obtained by Falgout, Vassilevski and Zikatanov (Numer. Linear Algebra Appl. 2005; 12:471–494), and also previous algebraic bounds, which can be seen as corollaries of this estimate. These results allow to measure the convergence by checking ‘approximation properties’. In this work, proper extensions of the latter to the nonsymmetric case are presented. Sometimes approximation properties for the SPD case are summarized in loose terms; e.g.: Interpolation must be able to approximate an eigenvector with error bound proportional to the size of the eigenvalue (SIAM J. Sci. Comp. 2000; 22:1570–1592). It is shown that this can be applied to nonsymmetric problems too, understanding ‘size’ as ‘modulus’. Eventually, an analysis is developed, for the nonsymmetric case, of the theoretical foundations of ‘compatible relaxation’, according to which a Fine/Coarse partitioning may be checked and possibly improved. Copyright © 2009 John Wiley & Sons, Ltd.