This chapter explains the need for mixed finite element methods and the algorithmic ingredients of this discretization approach. Various Diffpack tools for easy programming of mixed methods on unstructured grids in 2D and 3D are described. As model problems for exemplifying the formulation and implementation of mixed finite elements we address the Stokes problem for creeping viscous flow and the system formulation of the Poisson equation. Efficient solution of the linear systems arising from mixed finite elements is treated in the chapter on block preconditioning.


Scalar Field Stokes Problem Mixed Finite Element Finite Element Space Isoparametric Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. N. Arnold, F. Brezzi, and M. Fortin. A stable finite element method for the stokes equations. Calcolo, 21:337–344, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, 1994.Google Scholar
  3. 3.
    F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, 1991.Google Scholar
  4. 4.
    A. M. Bruaset and H. P. Langtangen. A comprehensive set of tools for solving partial differential equations; Diffpack. In M. Daehlen and A. Tveito, editors, Mathematical Models and Software Tools in Industrial Mathematics, pages 61–90. Birkhäuser, 1997.Google Scholar
  5. 5.
    M. Crouzeix and P.A. Raviart. Conforming and non-conforming finite element methods for solving the stationary stokes equations. RAIRO Anal. Numér, 7:33–76, 1973.MathSciNetGoogle Scholar
  6. 6.
    R. S. Falk and J. E. Osborn. Error estimates for mixed methods. R.A.I.R.O. Numerical Analysis, 14:249–277, 1980.MathSciNetzbMATHGoogle Scholar
  7. 7.
    C. A. J. Fletcher. Computational Techniques for Fluid Dynamics, Vol I and II. Springer Series in Computational Physics. Springer-Verlag, 1988.Google Scholar
  8. 8.
    L. P. Franca, T. J. R. Hughes, and R. Stenberg. Stabilized finite element methods. In M. D. Gunzburger and R. A. Nicolaides, editors, Incompressible Computational Fluid Dynamics; Trends and Advances. Cambridge University Press, 1993.Google Scholar
  9. 9.
    V. Girault and P. A. Raviart. Finite Element Methods for Navier-Stokes Equa-tions. Springer-Verlag, 1986.Google Scholar
  10. 10.
    C. Johnson. Numerical solution of partial differential equations by the finite element method. Studentlitteratur, 1987.Google Scholar
  11. 11.
    H. P. Langtangen. Tips and frequently asked questions about Diffpack. World Wide Web document: Diffpack v1.4 Report Series, SINTEF & University of Oslo, 1996. Google Scholar
  12. 12.
    H. P. Langtangen. Computational Partial Differential Equations-Numerical Methods and Diffpack Programming. Textbooks in Computational Science and Engineering. Springer, 2nd edition, 2003.Google Scholar
  13. 13.
    H. P. Langtangen. Details of finite element programming in Diffpack. The Numerical Objects Report Series #1997:9, Numerical Objects AS, Oslo, Norway, October 6, 1997. See Scholar
  14. 14.
    K.-A. Mardal, H. P. Langtangen, and G.W. Zumbusch. Software tools for multigrid methods. In H. P. Langtangen and A. Tveito, editors, Advanced Topics in Computational Partial Differential Equations-Numerical Methods and Diffpack Programming. Springer, 2003.Google Scholar
  15. 15.
    K.-A. Mardal, J. Sundnes, H.P Langtangen, and A. Tveito. Systems of PDEs and block preconditionering. In H. P. Langtangen and A. Tveito, editors, Advanced Topics in Computational Partial Differential Equations-Numerical Methods and Diffpack Programming. Springer, 2003.Google Scholar
  16. 16.
    K.-A. Mardal, X.-C. Tai, and R. Winther. Robust finite elements for Darcy-Stokes flow. SIAM Journal on Numerical Analysis, 40:1605–1631, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    R. Rannacher. Finite element methods for the incompressible Navier-Stokes equation. 1999. 1999.html. Google Scholar
  18. 18.
    R. Rannacher and S. Turek. A simple nonconforming quadrilateral stokes element. Numer. Meth. Part. Diff. Equ, 8:97–111, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    P. A. Raviart. Mixed finite element methods. In D. F. Griffiths, editor, The Mathematical Basis of Finite Element Methods. Clarendon Press, Oxford, 1984.Google Scholar
  20. 20.
    P. A. Raviart and J. M. Thomas. A mixed finite element method for 2-order elliptic problems. Matematical Aspects of Finite Element Methods, 1977.Google Scholar
  21. 21.
    S. Turek. Efficient Solvers for Incompressible Flow Problems. Springer, 1999.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • K.-A. Mardal
    • 1
    • 2
  • H. P. Langtangen
    • 1
    • 2
  1. 1.Simula Research LaboratoryNorway
  2. 2.Department of InformaticsUniversity of OsloNorway

Personalised recommendations