Adaptive algebraic multigrid

Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limit...

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Bibliographic Details
Published in:SIAM journal on scientific computing 2006, Vol.27 (4), p.1261-1286
Main Authors: BREZINA, M, FALGOUT, R, MACLACHLAN, S, MANTEUFFEL, T, MCCORMICK, S, RUGE, J
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Applied mathematics
Eigenvectors
Exact sciences and technology
Iterative methods
Laboratories
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Partial differential equations, boundary value problems
Sciences and techniques of general use
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Summary:Efficient numerical simulation of physical processes is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to unsatisfied assumptions made on the near null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. Emphasis is on the principles that guide the adaptivity and their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
ISSN:1064-8275
1095-7197
DOI:10.1137/040614402