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Variable-step multilevel preconditioning methods, I: Self-adjoint and positive definite elliptic problems (1994)

Axelsson, O. ; Vassilevski, P. S.

New York, NY [u.a.] : Wiley-Blackwell

Numerical Linear Algebra with Applications 1 (1994), S. 75-101

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Details

ISSN: 1070-5325

Schlagwort(e): Variable-step preconditioners ; Nonlinear preconditioning ; Generalized conjugate gradient method ; Engineering ; Engineering General

Quelle: Wiley InterScience Backfile Collection 1832-2000

Thema: Mathematik

Notizen: When solving large size systems of equations by preconditioned iterative solution methods, one normally uses a fixed preconditioner which may be defined by some eigenvalue information, such as in a Chebyshev iteration method. In many problems, however, it may be more effective to use variable preconditioners, in particular when the eigenvalue information is not available.In the present paper, a recursive way of constructing variable-step of, in general, nonlinear multilevel preconditioners for selfadjoint and coercive second-order elliptic problems, discretized by the finite element method is proposed. The preconditioner is constructed recursively from the coarsest to finer and finer levels. Each preconditioning step requires only block-diagonal solvers at all levels except at every k0, k0 ≥ 1 level where we perform a sufficient number ν, ν ≥ 1 of GCG-type variable-step iterations that involve the use again of a variable-step preconditioning for that level.It turns out that for any sufficiently large value of k0 and, asymptotically, for ν sufficiently large, but not too large, the method has both an optimal rate of convergence and an optimal order of computational complexity, both for two and three space dimensional problem domains.The method requires no parameter estimates and the convergence results do not depend on the regularity of the elliptic problem.

Zusätzliches Material: 8 Tab.

Materialart: Digitale Medien

URL: http://dx.doi.org/10.1002/nla.1680010108

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