A Smoother Based on Nonoverlapping Domain Decomposition Methods for H(div) Problems: A Numerical Study

  • Susanne C. BrennerEmail author
  • Duk-Soon OhEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)


The purpose of this paper is to introduce a V-cycle multigrid method for vector field problems discretized by the lowest order Raviart-Thomas hexahedral element. Our method is connected with a smoother based on a nonoverlapping domain decomposition method. We present numerical experiments to show the effectiveness of our method.


  1. 1.
    D.N. Arnold, R.S. Falk, R. Winther, Preconditioning discrete approximations of the Reissner-Mindlin plate model. RAIRO Modél. Math. Anal. Numér. 31(4), 517–557 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    D.N. Arnold, R.S. Falk, R. Winther, Preconditioning in H(div) and applications. Math. Comput. 66(219), 957–984 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.N. Arnold, R.S. Falk, R. Winther, Multigrid preconditioning in H(div) on non-convex polygons. Comput. Appl. Math. 17(3), 303–315 (1998)MathSciNetzbMATHGoogle Scholar
  4. 4.
    D.N. Arnold, R.S. Falk, R. Winther, Multigrid in H(div) and H(curl). Numer. Math. 85(2), 197–217 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S.C. Brenner, D.-S. Oh, Multigrid methods for H(div) in three dimensions with nonoverlapping domain decomposition smoothers. Numer. Linear Algebra Appl.
  6. 6.
    Z. Cai, R.D. Lazarov, T.A. Manteuffel, S.F. McCormick, First-order system least squares for second-order partial differential equations: part I. SIAM J. Numer. Anal. 31(6), 1785–1799 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Girault, P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5 (Springer, Berlin, 1986)Google Scholar
  8. 8.
    R. Hiptmair, Multigrid method for H(div) in three dimensions. Electron. Trans. Numer. Anal. 6(Dec), 133–152 (1997), Special issue on multilevel methods (Copper Mountain, CO, 1997)Google Scholar
  9. 9.
    R. Hiptmair, A. Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, in Parallel Solution of Partial Differential Equations (Minneapolis, MN, 1997). IMA Volumes in Mathematics and Its Applications, vol. 120 (Springer, New York, 2000), pp. 181–208Google Scholar
  10. 10.
    T.V. Kolev, P.S. Vassilevski, Parallel auxiliary space AMG solver for H(div) problems. SIAM J. Sci. Comput. 34(6), A3079–A3098 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    K.-A. Mardal, R. Winther, Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18(1), 1–40 (2011)MathSciNetCrossRefGoogle Scholar
  12. 12.
    D.-S. Oh, An overlapping Schwarz algorithm for Raviart-Thomas vector fields with discontinuous coefficients. SIAM J. Numer. Anal. 51(1), 297–321 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D.-S. Oh, O.B. Widlund, S. Zampini, C.R. Dohrmann, BDDC algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart–Thomas vector fields. Math. Comput. 87, 659–692 (2018) MathSciNetCrossRefGoogle Scholar
  14. 14.
    P.-A. Raviart, J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Lecture Notes in Mathematics, vol. 606 (Springer, Berlin, 1977), pp. 292–315Google Scholar
  15. 15.
    P.S. Vassilevski, U. Villa, A block-diagonal algebraic multigrid preconditioner for the Brinkman problem. SIAM J. Sci. Comput. 35(5), S3–S17 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    B.I. Wohlmuth, A. Toselli, O.B. Widlund, An iterative substructuring method for Raviart-Thomas vector fields in three dimensions. SIAM J. Numer. Anal. 37(5), 1657–1676 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

Personalised recommendations