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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Parallel multilevel preconditioners
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by James H. Bramble, Joseph E. Pasciak and Jinchao Xu PDF
Math. Comp. 55 (1990), 1-22 Request permission

Abstract:

In this paper, we provide techniques for the development and analysis of parallel multilevel preconditioners for the discrete systems which arise in numerical approximation of symmetric elliptic boundary value problems. These preconditioners are defined as a sum of independent operators on a sequence of nested subspaces of the full approximation space. On a parallel computer, the evaluation of these operators and hence of the preconditioner on a given function can be computed concurrently. We shall study this new technique for developing preconditioners first in an abstract setting, next by considering applications to second-order elliptic problems, and finally by providing numerically computed condition numbers for the resulting preconditioned systems. The abstract theory gives estimates on the condition number in terms of three assumptions. These assumptions can be verified for quasi-uniform as well as refined meshes in any number of dimensions. Numerical results for the condition number of the preconditioned systems are provided for the new algorithms and compared with other well-known multilevel approaches.
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Math. Comp. 55 (1990), 1-22
  • MSC: Primary 65N30; Secondary 65F10
  • DOI: https://doi.org/10.1090/S0025-5718-1990-1023042-6
  • MathSciNet review: 1023042