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Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic ProblemsMultigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.
Document ID
19970006617
Acquisition Source
Langley Research Center
Document Type
Conference Paper
Authors
Vanek, Petr
(Colorado Univ. Denver, CO United States)
Mandel, Jan
(Colorado Univ. Denver, CO United States)
Brezina, Marian
(Colorado Univ. Denver, CO United States)
Date Acquired
August 17, 2013
Publication Date
September 1, 1996
Publication Information
Publication: Seventh Copper Mountain Conference on Multigrid Methods
Subject Category
Numerical Analysis
Accession Number
97N13546
Funding Number(s)
CONTRACT_GRANT: NSF ASC-92-17394
CONTRACT_GRANT: NSF ASC-91-21431
Distribution Limits
Public
Copyright
Work of the US Gov. Public Use Permitted.
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