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Variable‐step multilevel preconditioning methods, I: Self‐adjoint and positive definite elliptic problems
Author(s) -
Axelsson O.,
Vassilevski P. S.
Publication year - 1994
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.1680010108
Subject(s) - preconditioner , mathematics , discretization , eigenvalues and eigenvectors , condition number , variable (mathematics) , rate of convergence , convergence (economics) , iterative method , chebyshev filter , positive definite matrix , finite element method , elliptic curve , nonlinear system , mathematical optimization , mathematical analysis , computer science , channel (broadcasting) , physics , computer network , quantum mechanics , economics , thermodynamics , economic growth
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 k 0 , k 0 ≥ 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 k 0 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.