Parallel multilevel preconditioners

James H. Bramble, Joseph E. Pasciak, Jinchao Xu

Research output: Contribution to journalArticle

383 Citations (Scopus)

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 wellknown multilevel approaches.

Original languageEnglish (US)
Pages (from-to)1-22
Number of pages22
JournalMathematics of Computation
Volume55
Issue number191
DOIs
StatePublished - Jul 1990

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Multilevel Preconditioners
Condition number
Preconditioner
Boundary value problems
Second-order Elliptic Problems
Approximation Space
Elliptic Boundary Value Problems
Parallel Computers
Operator
Numerical Approximation
Discrete Systems
Subspace
Mesh
Numerical Results
Evaluation
Estimate

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

Cite this

Bramble, James H. ; Pasciak, Joseph E. ; Xu, Jinchao. / Parallel multilevel preconditioners. In: Mathematics of Computation. 1990 ; Vol. 55, No. 191. pp. 1-22.
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Parallel multilevel preconditioners. / Bramble, James H.; Pasciak, Joseph E.; Xu, Jinchao.

In: Mathematics of Computation, Vol. 55, No. 191, 07.1990, p. 1-22.

Research output: Contribution to journalArticle

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