Parallel Application of a Novel Domain Decomposition Preconditioner for the Stable Finite Element Solution of Three-Dimensional Convection-Dominated PDEs

  • Peter K. Jimack
  • Sarfraz A. Nadeem
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2150)

Abstract

We describe and analyze the parallel implementation of a novel domain decomposition preconditioner for the fast iterative solution of linear systems of algebraic equations arising from the discretization of elliptic partial differential equations (PDEs) in three dimensions. In previous theoretical work, [3], this preconditioner has been proved to be optimal for symmetric positive-definite (SPD) linear systems. In this paper we provide details of our 3-d parallel implementation and demonstrate that the technique may be generalized to the solution of non-symmetric algebraic systems, such as those arising when convection-diffusion problems are discretized using either Galerkin or stabilized finite element methods (FEMs), [9].

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ashby, S. F., Manteuffel, T. A., Taylor, P. E.: A Taxonomy for Conjugate Gradient Methods. SIAM J. on Numer. Anal. 27 (1990) 1542–1568.MATHCrossRefGoogle Scholar
  2. 2.
    Bank, R. E., Jimack, P. K.: A New Parallel Domain Decomposition Method for the Adaptive Finite Element Solution of Elliptic Partial Differential Equations. Concurrency and Computation: Practice and Experience 13 (2001) 327–350.MATHCrossRefGoogle Scholar
  3. 3.
    Bank, R. E., Jimack, P. K., Nadeem, S. A., Nepomnyaschikh, S. V.: A Weakly Overlapping Domain Decomposition for the Adaptive Finite Element Solution of Elliptic Partial Differential Equations. Submitted to SIAM J. on Sci. Comp. (2001).Google Scholar
  4. 4.
    Cai, X.-C., Sarkis, M.: A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems. SIAM J. on Sci. Comp. 21 (1999) 792–797.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chan, T., Mathew, T.: Domain Decomposition Algorithms. Acta Numerica 3 (1994) 61–143.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Farhat, C., Mandel, J., Roux, F. X.: Optimal Convergence Properties of the FETI Domain Decomposition Method. Comp. Meth. for Appl. Mech. and Eng. 115 (1994) 365–385.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gropp, W. D., Keyes, D. E.: Parallel Performance of Domain-Decomposed Preconditioned Krylov Methods for PDEs with Locally Uniform Refinement. SIAM J. on Sci. Comp. 13 (1992) 128–145.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hodgson, D. C., Jimack, P. K.: A Domain Decomposition Preconditioner for a Parallel Finite Element Solver on Distributed Unstructured Grids. Parallel Computing 23 (1997) 1157–1181.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Johnson, C.: Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press (1987).Google Scholar
  10. 10.
    Message Passing Interface Forum: MPI: A Message-Passing Interface Standard. Int. J. Supercomputer Appl. 8 (1994) no. 3/4.Google Scholar
  11. 11.
    Oswald, P.: Multilevel Finite Element Approximation: Theory and Applications. Teubner Skripten zur Numerik, B. G. Teubner (1994).Google Scholar
  12. 12.
    Saad, Y.: SPARSEKIT: A Basic Tool Kit for Sparse Matrix Computations, Version 2. Technical Report, Center for Supercomputing Research and Development, University of Illinois at Urbana-Champaign, Urbana, IL, USA (1994).Google Scholar
  13. 13.
    Smith, B., Bjorstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996).Google Scholar
  14. 14.
    Speares, W., Berzins, M.: A 3-D Unstructured Mesh Adaptation Algorithm for Time-Dependent Shock Dominated Problems. Int. J. for Numer. Meth. in Fluids 25 (1997) 81–104.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Peter K. Jimack
    • 1
  • Sarfraz A. Nadeem
    • 1
  1. 1.Computational PDEs Unit, School of ComputingUniversity of LeedsLeedsUK

Personalised recommendations