Algebraic Multilevel Preconditioning Methods, II

In a previous paper Axelsson and Vassilevski presented a multilevel method to solve second-order self-adjoint elliptic problems which has an optimal order of computational complexity under mild restrictions of the finite-element mesh if the coefficients are smooth within elements of the coarsest ini...

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Vydáno v:SIAM journal on numerical analysis Ročník 27; číslo 6; s. 1569 - 1590
Hlavní autoři: Axelsson, O., Vassilevski, P. S.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.12.1990
Témata:
Algebra
Algorithms
Applied mathematics
Approximation
Exact sciences and technology
Factorization
Informatics
Iterative methods
Mathematics
Matrices
Methods
Multigrid methods
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Polynomials
Preconditioning
Sciences and techniques of general use
Triangulation
Finite element method
Second order equation
Elliptic equation
Approximate method
Multilevel method
Recursive method
Matrix decomposition
Schur complement
Block matrix
Preconditioning
Computing complexity
ISSN:0036-1429, 1095-7170
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Shrnutí:In a previous paper Axelsson and Vassilevski presented a multilevel method to solve second-order self-adjoint elliptic problems which has an optimal order of computational complexity under mild restrictions of the finite-element mesh if the coefficients are smooth within elements of the coarsest initial mesh. The preconditioner is based on an approximate factorization of matrices partitioned in two-by-two block form where the inverses of the arising Schur complements are approximated by certain matrix polynomials, involving the inverse of the preconditioner on the next coarser mesh level, and the Schur complement itself (version (i)) or the stiffness matrix on the next coarser level (version (ii)). In this way the preconditioner is defined recursively from one level to the next until the coarsest mesh, where it is assumed that the matrix is solved by a direct solution method, for instance. There is a relation,$\nu > (1 - \gamma^2)^{-1/2}$, between the degree (ν) of these polynomials and the constant (γ) in the method for the fundamental inequality, the extended Cauchy-Bunyakowski-Schwarz (CBS) inequality for the elliptic inner product and corresponding hierarchical finite-element basis functions, and this implies an optimal rate of convergence of the corresponding iterative method. In the first paper it was assumed that the systems to be eliminated on each level containing the major block matrices (denoted A(k) 11), of the given matrix were solved exactly. In the present paper an implementation of version (ii) is considered, where this matrix is also approximated. It is shown that the same relation between nu and γ holds for the extended version also, for an optimal rate of convergence.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0036-1429
1095-7170
DOI:10.1137/0727092