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Mathematics of Computation

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



Parallel multilevel preconditioners

Authors: James H. Bramble, Joseph E. Pasciak and Jinchao Xu
Journal: Math. Comp. 55 (1990), 1-22
MSC: Primary 65N30; Secondary 65F10
MathSciNet review: 1023042
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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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Article copyright: © Copyright 1990 American Mathematical Society