MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian



Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation.


Saddle point system Multigrid methods Mixed finite elements Vector Laplacian Maxwell equations 

Mathematics Subject Classification

65N55 65F10 65N22 65N30 


  1. 1.
    Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arnold, D.N., Falk, R.S., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85, 197–218 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bramble, J.H., Pasciak, J.E., Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comput. 56, 1–34 (1991)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bramble, J.H., Pasciak, J.E.: The analysis of smoothers for multigrid algorithms. Math. Comput. 58, 467–488 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brezzi, F., Douglas, J., Duran, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  9. 9.
    Chen, L.: \(i\)FEM: an integrated finite element methods package in matlab, Technical report, University of California at Irvine (2009)Google Scholar
  10. 10.
    Chen, L., Wang, M., Zhong, L.: Convergence Analysis of Triangular MAC Schemes for Two Dimensional Stokes Equations. J. Sci. Comput.
  11. 11.
    Chen, L., Wu, Y.: Convergence of adaptive mixed finite element methods for the Hodge Laplacian equations: without harmonic forms. SIAM J. Numer. Anal. 55(6), 2905–2929 (2017)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chen, L., Wu, Y.: Convergence Analysis for A Class of Iterative Methods for Solving Saddle Point Systems, arXiv:1710.03409 [math.NA]
  13. 13.
    Chen, J., Xu, Y., Zou, J.: An adaptive inverse iteration for Maxwell eigenvalue problem based on edge elements. J. Comput. Phys. 229(7), 2649–2658 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ciarlet, P., Wu, J,H., Zou, J.: Edge element methods for maxwells equations with strong convergence for gauss laws. SIAM J. Numer. Anal. 53(4), 2350–2372 (2015)Google Scholar
  15. 15.
    Elman, H. C.: Iterative Methods for Large Sparse Non-Symmetric Systems of Linear Equations, Ph.D. thesis, Yale University, New Haven, CT (1982)Google Scholar
  16. 16.
    Fortin, M., Glowinski, R.: Augmented Lagrangian Methods, Applications to the Numerical Solution of Boundary Value Problems. North-Holland Publishing Co., Amsterdam (1983)MATHGoogle Scholar
  17. 17.
    Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations. Springer, New York (1986)CrossRefMATHGoogle Scholar
  18. 18.
    Hiptmair, R.: Multigrid method for H(div) in three dimensions. Electron. Trans. Numer. Anal. 6, 133–152 (1997)MathSciNetMATHGoogle Scholar
  19. 19.
    Hiptmair, R.: Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36(1), 204–225 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hiptmair, R., Xu, J.: Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45(6), 2483–2509 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Loghin, D., Wathen, A.J.: Analysis of preconditioners for saddle-point problems. SIAM J. Sci. Comput. 25, 20292049 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Mardal, K.A., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98, 305–327 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Monk, P.: Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29, 714–729 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRefMATHGoogle Scholar
  26. 26.
    Nédélec, J.C.: Mixed finite elements in \(R^{3}\). Numer. Math. 35, 315–341 (1980)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nédélec, J.C.: A new family of mixed finite elements in \(R^{3}\). Numer. Math. 50, 57–81 (1986)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Olshanskii, Maxim A., Tyrtyshnikov, Eugene E.: Iterative Methods for Linear Systems Theory and Applications Society for Industrial and Applied Mathematics, Philadelphia (2014)Google Scholar
  29. 29.
    Raviart, P.A., Thomas, J.: A mixed finite element method fo 2-nd order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical aspects of the Finite Elements Method. Lectures Notes in Math, pp. 292–315. Springer, Berlin (1977)Google Scholar
  30. 30.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS, Boston (1996)MATHGoogle Scholar
  31. 31.
    Zhou, J., Hu, X., Zhong, L., Shu, S., Chen, L.: Two-grid methods for maxwell eigenvalue problem. SIAM J. Numer. Anal. 52(4), 2027–2047 (2014)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Beijing Institute for Scientific and Engineering ComputingBeijing University of TechnologyBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduChina
  4. 4.School of Mathematical and Computational SciencesXiangtan UniversityXiangtanChina

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