Solving the Stokes problem on a massively parallel computer

Owe Axelsson, Vincent A. Barker, Maya Neytcheva, B. Polman

    Research output: Contribution to journalJournal articleResearchpeer-review

    Abstract

    We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.
    Original languageEnglish
    JournalMathematical Modelling and Analysis
    Volume6
    Issue number1
    Pages (from-to)7-27
    ISSN1392-6292
    DOIs
    Publication statusPublished - 2001

    Cite this

    Axelsson, Owe ; Barker, Vincent A. ; Neytcheva, Maya ; Polman, B. / Solving the Stokes problem on a massively parallel computer. In: Mathematical Modelling and Analysis. 2001 ; Vol. 6, No. 1. pp. 7-27.
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    abstract = "We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.",
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    Solving the Stokes problem on a massively parallel computer. / Axelsson, Owe; Barker, Vincent A.; Neytcheva, Maya; Polman, B.

    In: Mathematical Modelling and Analysis, Vol. 6, No. 1, 2001, p. 7-27.

    Research output: Contribution to journalJournal articleResearchpeer-review

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    T1 - Solving the Stokes problem on a massively parallel computer

    AU - Axelsson, Owe

    AU - Barker, Vincent A.

    AU - Neytcheva, Maya

    AU - Polman, B.

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    N2 - We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.

    AB - We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.

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