Auxiliary Space Preconditioners for Mixed Finite Element Methods

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 70)


This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular H(div) system. Numerical experiments also justify the robustness and efficiency of this scheme.


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  1. 1.
    Arnold, D.N., Falk, R.S., Winther, R.: Preconditioning in H (div) and applications. Math. Comp., 66:957–984, 1997.CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bank, R.E., Welfert, B.D., Yserentant, H.: A class of iterative methods for solving mixed finite element equations. Numer. Math., 56:643–666, 1990.MathSciNetGoogle Scholar
  3. 3.
    Beck, R.: Graph-based algebraic multigrid for Lagrange-type finite elements on simplicial. Technical Report Preprint SC 99-22, Konrad-Zuse-Zentrum für Informationstechnik Berlin, 1999.Google Scholar
  4. 4.
    Bochev, P.B., Seifert, C., Tuminaro, R., Xu, J., Zhu, Y.: Compatible gauge approaches for H(div) equations. In CSRI Summer Proceedings, 2007.Google Scholar
  5. 5.
    Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math., 47(2):217–235, 1985.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer, 1991.Google Scholar
  7. 7.
    Chen, Z., Ewing, R.E., Lazarov, R.D., Maliassov, S., Kuznetsov, Y.A.: Multilevel preconditioners for mixed methods for second order elliptic problems. Numer. Linear Algebra Appl., 3(5):427–453, 1996.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fortin, M., Glowinski, R. Augmented Lagrangian Methods: Application to the numerical solution of boundary value problems. North-Holland, Amsterdam, 1983.Google Scholar
  9. 9.
    Gee, M., Siefert, C., Hu, J., Tuminaro, R., Sala, M.: ML 5.0 smoothed aggregation user’s guide. Technical Report SAND2006-2649, Sandia National Laboratories, 2006.Google Scholar
  10. 10.
    Hiptmair, R.: Multigrid method for H(div) in three dimensions. Electron. Trans. Numer. Anal., 6:133–152, 1997.MATHMathSciNetGoogle Scholar
  11. 11.
    Hiptmair, R., Schiekofer, T., Wohlmuth, B. Multilevel preconditioned augmented Lagrangian techniques for 2nd order mixed problems. Computing, 57(1):25–48, 1996.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Hiptmair, R., Xu, J.: Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces. SIAM J. Numer. Anal., 45:2483, 2007.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kolev, T., Vassilevski, P.: Some experience with a H 1-based auxiliary space AMG for H(curl) problems. Technical Report UCRL-TR-221841, Lawrence Livermore Nat. Lab., 2006.Google Scholar
  14. 14.
    Lee, Y., Wu, J., Xu, J., Zikatanov, L. Robust subspace correction methods for nearly singular systems. Math. Models Methods Appl. Sci., 17(11):1937, 2007.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Raviart, P.A., Thomas, J.: A mixed finite element method fo 2nd order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical aspects of the Finite Elements Method, Lectures Notes in Math. 606, pages 292–315. Springer, Berlin, 1977.Google Scholar
  16. 16.
    Vaněk, P., Brezina, M., Mandel, J. Convergence of algebraic multigrid based on smoothed aggregation. Numer. Math., 88(3):559–579, 2001.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Vassilevski, P., Lazarov, R.: Preconditioning Mixed Finite Element Saddle-point Elliptic Problems. Numer. Linear Algebra Appl., 3(1):1–20, 1996.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Vassilevski, P.S., Wang, J.: Multilevel iterative methods for mixed finite element discretizations of elliptic problems. Numer. Math., 63(1):503–520, 1992.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Wohlmuth, B.I., Toselli, A., Widlund, O.B.: An iterative substructuring method for Raviart-Thomas vector fields in three dimensions. SIAM J. Numer. Anal., 37(5):1657–1676, 2000.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured meshes. Computing, 56:215–235, 1996.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.MS 9214, Sandia National LaboratoriesLivermoreUSA
  2. 2.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA
  3. 3.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

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