A Computational Comparison of Two FEM Solvers for Nonlinear Incompressible Flow

  • Jaroslav Hron
  • Abderrahim Ouazzi
  • Stefan Turek
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 35)


In this comparative study, we examine the influence of two different FEM discretization techniques (conforming Q2/Pinonconforming Q1/Qo Stokes element) and solution procedures (nonlinear Newton variants and multigrid vs. Krylov-space solvers for the linear subproblems) onto the approximation properties and particularly the total efficiency of corresponding CFD simulation tools. We discuss algorithmic details and give numerical results for laminar incompressible flow examples including non-Newtonian behavior ofpower law type.


Convective Term Lift Coefficient Total Efficiency Stabilization Technique Mesh Width 
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  1. 1.
    J. Baranger and K. Najib. Analyse numerique des écoulements quasi-newtoniens dont la viscosity obeit la loi puissance ou la loi de carreau.Numer. Math., 58:3549, 1990.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Bramley and X. Wang. SPLIB:A library of iterative methods for sparse linear systems. Department of Computer Science, Indiana University,Bloomington,IN,1997. Scholar
  3. 3.
    C. S. Brenner. Korn’s inequalities for piecewise H1vector fields.IMI Preprint Series, 5:1–21, 2002.Google Scholar
  4. 4.
    P. Hansbo and M. G. Larson. A simple nonconforming bilinear element for the elasticity problem. InCIMNEBarcelona, Spain, 2001.Google Scholar
  5. 5.
    J. Hron, J. Málek, J. Neéas, and K. R. Rajagopal. Numerical simulations and global existence of solutions of two dimensional flows of fluids with pressure and shear dependent viscosities.Math. Comp. Sim., 2002. to appear.Google Scholar
  6. 6.
    D. Kuzmin, M. Möller, and S. Turek. Multidimensional fem-fct schemes for arbitrary time-stepping.J. Comp. Phys., 2002. (to appear).Google Scholar
  7. 7.
    A. Ouazzi, R. Schmachtel, and S. Turek. Multigrid methods for stabilized nonconforming finite elements for incompressible flow involving the deformation tensor formulation.J. Numer. Math.10:235–248, 2002.MathSciNetzbMATHGoogle Scholar
  8. 8.
    A. Prohl and M. Rúziéka. On fully implicit space-time discretization for motions of incompressible fluids with shear-dependent viscosity: the case p < 2. SIAMJ. Numer. Anal.39:214–249, 2001.CrossRefzbMATHGoogle Scholar
  9. 9.
    R. Rannacher and S. Turek. A simple nonconforming quadrilateral stokes element.Numer. Meth. Part. Diff. Equ.8:97–111, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems.SIAM J. Sci. Stat. Comput.7:856–869, 1986.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Schaeffer. Instability in the evolution equations describing incompressible granular flow.J. Diff. Eq.66:19–50, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Schäfer and S. Turek. Benchmark computations of laminar flow around cylinder. In E.H. Hirschel, editorFlow Simulation with High-Performance Computers II, volume 52 of Notes on Numerical Fluid Mechanicspages 547–566. Vieweg, 1996.CrossRefGoogle Scholar
  13. 13.
    R. Schmachtel.Robuste lineare und nichtlineare Lösungsverfahren für die inkompressiblen Navier-Stokes-Gleichungen. PhD thesis, University of Dortmund, 2002. (to be published).Google Scholar
  14. 14.
    S. Turek.Efficient solvers for incompressible flow problems: An algorithmic and computational approachvolume 6 ofLNCSE.Springer, Berlin, 1999.Google Scholar
  15. 15.
    S. Turek et al. FEATFLOW -Finite element software for the incompressible Navier-Stokes equations: User Manual, Release1.2, 1999. Google Scholar
  16. 16.
    S. Turek and S. Kilian.The Virtual Album of Fluid Motion.Multimedia DVD. Springer, 2002. Scholar
  17. 17.
    H. A. van der Vorst. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems.SIAM J. Sci. Stat. Comput.13(2):631–644, 1992.CrossRefzbMATHGoogle Scholar
  18. 18.
    S. P. Vanka. Implicit multigrid solutions of navier-stokes equations in primitive variables.J. Comp. Phys.65:138–158, 1985.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jaroslav Hron
    • 1
  • Abderrahim Ouazzi
    • 1
  • Stefan Turek
    • 1
  1. 1.Institute of Applied MathematicsUniversity of DortmundDortmundGermany

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