Computational Geosciences

, Volume 14, Issue 2, pp 289–299 | Cite as

Multigrid preconditioned conjugate-gradient solver for mixed finite-element method

Open Access
Original paper


The mixed finite-element approximation to a second-order elliptic PDE results in a saddle-point problem and leads to an indefinite linear system of equations. The mixed system of equations can be transformed into coupled symmetric positive-definite matrix equations, or a Schur complement problem, using block Gauss elimination. A preconditioned conjugate-gradient algorithm is used for solving the Schur complement problem. The mixed finite-element method is closely related to the cell-centered finite difference scheme for solving second-order elliptic problems with variable coefficients. For the cell-centered finite difference scheme, a simple multigrid algorithm can be defined and used as a preconditioner. For distorted grids, an additional iteration is needed. Nested iteration with a multigrid preconditioned conjugate gradient inner iteration results in an effective numerical solution technique for the mixed system of linear equations arising from a discretization on distorted grids. Numerical results show that the preconditioned conjugate-gradient inner iteration is robust with respect to grid size and variability in the hydraulic conductivity tensor.


Mixed finite element method Lowest-order Raviart–Thomas Subsurface flow Multigrid Distorted grids Conjugate gradient Nested iteration 


  1. 1.
    Akin, J.E.: Application and Implementation of Finite Element Methods, pp. 153–158. Academic, London (1982)MATHGoogle Scholar
  2. 2.
    Allen, M.B., Ewing, R.E., Lu, P.: Well-conditioned iterative schemes for mixed finite-element models of porous-media flows. SIAM J. Sci. Statist. Comput. 13(3), 794–814 (1992)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, New York (1988)Google Scholar
  4. 4.
    Bank, R., Welfert, B., Yserentant, H.: A class of iterative methods for solving saddle point problems. Numer. Math. 55, 645–666 (1990)MathSciNetGoogle Scholar
  5. 5.
    Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., der Vorst, H.V.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994). Google Scholar
  6. 6.
    Bramble, J., Ewing, R., Pasciak, J., Shen, J.: The analysis of multigrid algorithms for cell centered finite difference methods. Adv. Comput. Math. 5(1), 15–29 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bramble, J., Pasciak, J., Apostol, T.: Analysis of the inexact uzawa algorithm for saddle point problems. SIAM J. Numer. Anal. 34(3), 1072–1092 (1987)CrossRefGoogle Scholar
  8. 8.
    Bramble, J., Pasciak, J., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comput. 56(193), 1–34 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brenner, S.C.: A mutligrid algorithm for the lowest-order Raviart–Thomas mixed triangular finite element method. SIAM J. Numer. Anal. 29(3), 647–678 (1992)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)MATHGoogle Scholar
  11. 11.
    Briggs, W.L.: A Multigrid Tutorial. Siam, Philadelphia (1987)MATHGoogle Scholar
  12. 12.
    Cai, Z., Jones, J., McCormick, S., Russell, T.: Control-volume mixed finite elements methods. Comput. Geosci. 1, 289–315 (1997)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chou, S., Kwak, D.Y., Kim, K.Y.: Flux recovery from primal hybrid finite element methods. SIAM J. Numer. Anal. 40(2), 403–415 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Dougherty, D.: PCG solutions of flow problems in random porous media using mixed finite elements. Adv. Water Resour. 13(1) (1990)Google Scholar
  15. 15.
    Ewing, R., Shen, J.: A multigrid algorithm for the cell-centered finite difference scheme. In: The Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods. NASA Conference Publication 3224 (1993)Google Scholar
  16. 16.
    Ewing, R., Wheeler, M.: Computational aspects of mixed finite element methods. In: Numerical Methods for Scientific Computing, pp. 163–172 (1983)Google Scholar
  17. 17.
    Golub, G.H., Van Loan, C.F.: Matrix Computations. John Hopkins University Press, Baltimore (1989)MATHGoogle Scholar
  18. 18.
    Harbaugh, A.W., Banta, E.R., Hill, M.C., McDonald, M.G.: Modflow-2000, the U.S. geological survey modular ground-water model user guide to modularization concepts and the ground-water flow process. Tech. rep., U.S. Geological Survey. Open-File Report 00-92 (2000)Google Scholar
  19. 19.
    Howard, C.E., Golub, G.H.: Inexact and preconditioned uzawa algorithms for saddle point problems. SIAM J. Numer. Anal. 31(6), 1645–1661 (1994)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hughes, T.: The Finite Element Method, pp. 123–125. Prentice-Hall, Englewood Cliffs (1987)MATHGoogle Scholar
  21. 21.
    Johnson, C.: Numerical Solutions of Partial Differential Equations by the Finite Element Method, pp. 141–144. Cambridge University Press, Cambridge (1987)Google Scholar
  22. 22.
    Naff, R., Russell, T., Wilson, J.: Test functions for three-dimensional control-volume finite-element methods on irregular grids. In: Computational Methods in Water Resources, vol. 2, pp. 677–684 (2000).
  23. 23.
    Naff, R., Russell, T., Wilson, J.: Shape functions for three-dimensional control-volume mixed finite-element methods on irregular grids. In: Computational Methods in Water Resources, pp. 359–366 (2002).
  24. 24.
    Naff, R., Russell, T.R., Wilson, J.: Shape functions for velocity interpolation in general hexahedral cells. Comput. Geosci. 6, 285–314 (2002). MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Raviart, P., Thomas, J.: A mixed finite element method for 2nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, pp. 292–315. Springer, New York (1977)CrossRefGoogle Scholar
  26. 26.
    Russell, T.F., Wheeler, M.F.: Finite element and finite difference methods for continuous flows in porous media. In: Ewing, R.E. (ed.) The Mathematics of Reservoir Simulation, pp. 35–106. Society of Industrial and Applied Mathematics, Philadelphia (1983)Google Scholar
  27. 27.
    Sameh, A., Baggag, A.: Nested iterative schemes for indefinite linear systems. In: Mang, H.A., Rammerstorfer, F.G., Eberhardsteiner, J. (eds.) Fifth World Congress on Computational Mechanics (2002)Google Scholar
  28. 28.
    Tong, Z., Sameh, A.: On an iterative method for saddle point problems. Numer. Math. 79, 643–646 (1998)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic, London (2001)MATHGoogle Scholar
  30. 30.
    Wilson, J., Naff, R.: Modflow-2000, the U.S. Geological Survey modular ground-water model – GMG linear equation solver package documentation. Tech. rep., U.S. Geological Survey (2004).
  31. 31.
    Xu, J.: Iterative methods by space decomposition and subspace corrections. SIAM Rev. 34(4), 581–613 (1992)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.U.S. Geological SurveyDenver Federal CenterDenverUSA

Personalised recommendations