Parallel Finite Element Methods for the Incompressible Navier-Stokes Equations

  • O. Dorok
  • V. John
  • U. Risch
  • F. Schieweck
  • L. Tobiska
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 48)


We consider parallel and adaptive algorithms for the incompressible Navier-Stokes equations discretized by an upwind type finite element method. Two parallelization concepts are used, a first one based on a static domain decomposition into macroelements and a second one based on a dynamic load balancing strategy. We investigate questions of the scalability up to the massive parallel case and the use of a posteriori error estimators. The arising discrete systems are solved by parallelized multigrid methods which are applied either directly to the coupled system or within a projection method.


Coarse Grid Multigrid Method Posteriori Error Estimate Posteriori Error Estimator Convection Diffusion Problem 
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  1. [1]
    P. Bastian, Parallele adaptive Mehrgitterverfahren. PhD thesis, Universität Heidelberg, 1994.Google Scholar
  2. [2]
    M. Crouzeix, P.-A. Raviart, Conforming and nonconforming Finite Element Methods for solving the stationary Stokes equations. RAIRO Anal Numer. 7, 33–76 (1973).MathSciNetGoogle Scholar
  3. [3]
    O. Dorok, G. Lube, U. Risch, F. Schieweck, L. Tobiska, Finite Element discretization of the Navier-Stokes equations. In: E.H. Hirschel (Ed.): Flow Simulation with High-Performance Computers I, Notes on Numerical Fluid Mechanics, Vol. 38, Vieweg-Verlag 1993, pp. 67–78.Google Scholar
  4. [4]
    K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Introduction to adaptive methods for differential equations. Acta Numerica, 1–54 (1995).Google Scholar
  5. [5]
    V. John, A comparison of some error estimators for convection diffusion problems on a parallel computer. Preprint 12/94, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik, 1994.Google Scholar
  6. [6]
    C. Johnson, On computability and error control in CFD. Int. Jour. Num. Meth. Fluids 20, 777–788 (1995).zbMATHCrossRefGoogle Scholar
  7. [7]
    H. Reichert, G. Wittum, Solving the Navier-Stokes equations on unstructured grids. In: E.H. Hirschel (Ed.): Flow Simulation with High-Performance Computers I, Notes on Numerical Fluid Mechanics, Vol. 38, Vieweg-Verlag 1993, pp. 321–333.Google Scholar
  8. [8]
    H. Rentz-Reichert, G. Wittum, A comparison of smoothers and numbering strategies for laminar flow around a cylinder. In this publication.Google Scholar
  9. [9]
    R. Rannacher, S. Turek, Simple Nonconforming Quadrilateral Stokes Element. Numer. Methods Partial Differential Equations 8, 97–111 (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    F. Schieweck, A parallel multigrid algorithm for solving the Navier-Stokes equations. IMPACT Comput. Sci. Engrg. 5, 345–378 (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    F. Schieweck, Multigrid Convergence Rates of a Sequential and a Parallel Navier-Stokes Solver. In: Hackbusch, W., Wittum, G. (Eds.): Fast Solvers for Flow Problems, Notes on Numerical Fluid Mechanics, Vol.49, Vieweg-Verlag 1995, pp. 251–262.Google Scholar
  12. [12]
    F. Schieweck, L. Tobiska, A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation. RAIRO Model. Math. Anal. Numér. 23, 627–647 (1989).MathSciNetzbMATHGoogle Scholar
  13. [13]
    L. Tobiska, A note on the artificial viscosity of numerical schemes. Comp. Fluid Dynamics 5, 281–290 (1995).MathSciNetCrossRefGoogle Scholar
  14. [14]
    St. Turek, A comparative study of time stepping techniques for the incompressible Navier-Stokes equations: From fully implicit nonlinear schemes to semi-implicit projection methods, to appear, 1995.Google Scholar
  15. [15]
    S. Vanka, Block-implicit multigrid calculation of two-dimensional recirculating flows. Comput. Methods Appl. Mech. Engrg. 59, 29–48 (1986).zbMATHCrossRefGoogle Scholar
  16. [16]
    R. Verfürth, A posteriori error estimators for the Stokes Stokes equations. II Nonconforming discretizations. Numer. Math. 60, 235–249 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Bericht 180, Ruhr-Universität Bochum, Fakultät für Mathematik, 1995.Google Scholar

Copyright information

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

Authors and Affiliations

  • O. Dorok
    • 1
  • V. John
    • 1
  • U. Risch
    • 1
  • F. Schieweck
    • 1
  • L. Tobiska
    • 1
  1. 1.Institut für Analysis und NumerikOtto-von-Guericke Universität MagdeburgMagdeburgGermany

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