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On block preconditioners for saddle point problems with singular or indefinite (1, 1) block
Author(s) -
Krzyżanowski Piotr
Publication year - 2011
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.717
Subject(s) - mathematics , saddle point , block matrix , discretization , block (permutation group theory) , saddle , rate of convergence , matrix (chemical analysis) , iterative method , diagonal , conjugate gradient method , mathematical analysis , mathematical optimization , combinatorics , geometry , computer science , channel (broadcasting) , eigenvalues and eigenvectors , physics , computer network , materials science , quantum mechanics , composite material
Abstract We discuss a class of preconditioning methods for the iterative solution of symmetric algebraic saddle point problems, where the (1, 1) block matrix may be indefinite or singular. Such problems may arise, e.g. from discrete approximations of certain partial differential equations, such as the Maxwell time harmonic equations. We prove that, under mild assumptions on the underlying problem, a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) can be chosen in a way which guarantees that the convergence rate of the preconditioned conjugate residuals method is independent of the discretization mesh parameter. We provide examples of such preconditioners that do not require additional scaling. Copyright © 2010 John Wiley & Sons, Ltd.