On a Time and Space Parallel Multi-Grid Method Including Remarks on Filtering Techniques

  • Jens Burmeister
  • Wolfgang Hackbusch
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 48)


In this paper we discuss the numerical treatment of parabolic problems by multi-grid methods under the aspect of parallelisation. Reflecting the concept to treat the time and space variables independently, the time-parallel multi-grid method is combined with a space-parallel multi-grid method. The space-parallel multi-grid method could be interpreted as a global multi-grid with a special domain decomposition smoother. The smoother requires approximations for the SCHUR complement. This question leads to the discussion of filtering techniques in a more general situation. Adaptivity in time is achieved by using extrapolation techniques which offers a third source of parallelism in addition to the time and space parallelism.


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  1. [1]
    AUZINGER, W., FRANK, R., MACSEK, F.: ”Asymptotic Error Expansion for Stiff Equations: The implicit Euler Scheme”, SIAM J. of Num. Anal. 27 No. 1 (1990) pp. 66–104.MathSciNetGoogle Scholar
  2. [2]
    BASTIAN, P., BURMEISTER, J., HORTON, G.: ”Implementation of a parallel multi-grid method for parabolic partial differential equations” (1991) pp. 18–27 in [9].Google Scholar
  3. [3]
    BURMEISTER, J., HORTON, G.: ”Time-parallel multi-grid solution of the Navier-Stokes equations”, (1991) pp. 155–166 in [10].Google Scholar
  4. [4]
    BURMEISTER, J.: ”Time-Parallei Multi-Grid Methods”, (1993) pp. 56–66 in [14].Google Scholar
  5. [5]
    BURMEISTER, J, PAUL, R.: ”Time-adaptive solution of discrete parabolic problems with time-parallel multi-grid methods”, (1995) pp. 49–58 in [21].Google Scholar
  6. [6]
    CHAN, T.F., MATHEW, T.P.: ”Domain decomposition algorithms”, Acta Numerica (1994) pp. 61–143, Cambridge University Press.Google Scholar
  7. [7]
    DOROK, O., JOHN, V., RISCH, U., SCHIEWECK, F., TOBISKA, L.: ”Parallel Finite Element Methods for the Incompressible Navier-Stokes Equations”, in this publication.Google Scholar
  8. [8]
    GLOWINSKI, R., LIONS, J.-R. (Editors): ”Computing methods in applied sciences and engineering”, VI. Proc. of the 6th International Symposium on Comp. Methods in Applied Sciences and Engineering. Versaille, France, Dec. 12–16, 1983, North Holland, 1984.Google Scholar
  9. [9]
    HACKBUSCH, W. (Editor): ”Parallel Algorithms for Partial Differential Equations”, Proceedings of the Sixth GAMM-Seminar, Kiel, January 19–21, 1990, Notes on Numerical Fluid Mechanics, Volume 31, Vieweg-Verlag, Braunschweig, 1991.Google Scholar
  10. [10]
    HACKBUSCH, W., TROTTENBERG, U. (Editors): ”Multi-grid Methods III”, Proceedings of the 3rd European Conference on Multi-grid Methods, Bonn, October 1–4, 1990, International Series of Numerical Mathematics, Vol. 98, Birkhäuser Verlag, Basel, 1991.Google Scholar
  11. [11]
    HACKBUSCH, W.: ”Parabolic multi-grid methods”, (1984) in [8].Google Scholar
  12. [12]
    HACKBUSCH, W.: ”Multi-Grid Methods and Applications”, Springer Series in Computational Mathematics 4, Springer-Verlag, Berlin, Heidelberg, 1985.Google Scholar
  13. [13]
    HACKBUSCH, W.: ”Iterative Solution of Large Sparse Systems of Equations”, Applied Mathematical Sciences 95, Springer-Verlag, New York, 1993.Google Scholar
  14. [14]
    HIRSCHEL, E.H. (Editor): ”Flow Simulation with High-Performance Computers I”, Notes on Numerical Fluid Mechanics, Volume 38, Vieweg-Verlag, Braunschweig, 1993.Google Scholar
  15. [15]
    HORTON, G.: ”Ein zeitparalleles Lösungsverfahren für die Navier-Stokes-Gleichungen”, doctoral thesis, University of Erlangen-Nürnberg, 1991.Google Scholar
  16. [16]
    LILEK, Ž., PERIĆ, M., SCHRECK, E.: ”Parallelization of Implicit Methods for Flow Simulation”, (1993) pp. 135–146 in [21].Google Scholar
  17. [17]
    MEYER, A.: ”A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain”, Computing 45 (1990) pp. 217–234.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    RENTZ-REICHERT, H., WITTUM, G.: ”A Comparison of Smoothers and Numbering Strategies for Laminar Flow around a Cylinder”, in this publication.Google Scholar
  19. [19]
    THOMÉE, V.: ”Galerkin Finite Element Methods for Parabolic Problems”, Lecture Notes in Mathematics 1054, Springer-Verlag, Berlin, Heidelberg, 1984.Google Scholar
  20. [20]
    WAGNER, CH.: ”Frequenzfilternde Zerlegungen für unsymmetrische Matrizen und Matrizen mit stark variierenden Koeffizienten”, doctoral thesis, University of Stuttgart, 1995.Google Scholar
  21. [21]
    WAGNER, S. (Editor): ”Computational Fluid Dynamics on Parallel Systems”, Notes on Numerical Fluid Mechanics, Volume 50, Vieweg-Verlag, Braunschweig, 1995.Google Scholar
  22. [22]
    WITTUM, G.: ”Filternde Zerlegungen. Schnelle Löser für große Gleichungssysteme”, Teubner Skripten zur Numerik, B.G. Teubner Verlag, Stuttgart, 1992.Google Scholar

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© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1996

Authors and Affiliations

  • Jens Burmeister
    • 1
  • Wolfgang Hackbusch
    • 1
  1. 1.Mathematisches Seminar II Lehrstuhl Praktische MathematikUniversität KielKielGermany

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