Advertisement

Journal of Scientific Computing

, Volume 77, Issue 3, pp 1936–1952 | Cite as

Preconditioning of a Hybridized Discontinuous Galerkin Finite Element Method for the Stokes Equations

  • Sander Rhebergen
  • Garth N. Wells
Article
  • 140 Downloads

Abstract

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners.

Keywords

Stokes equations Preconditioning Hybridized methods Discontinuous Galerkin Finite element methods 

References

  1. 1.
    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc users manual. Technical Report ANL-95/11 - Revision 3.7, Argonne National Laboratory. http://www.mcs.anl.gov/petsc (2016)
  2. 2.
    Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc Web page. http://www.mcs.anl.gov/petsc (2016)
  3. 3.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Berlin (2013)zbMATHGoogle Scholar
  4. 4.
    Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)CrossRefGoogle Scholar
  5. 5.
    Cesmelioglu, A., Cockburn, B., Nguyen, N.C., Peraire, J.: Analysis of HDG methods for Oseen equations. J. Sci. Comput. 55, 392–431 (2013).  https://doi.org/10.1007/s10915-012-9639-y MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cesmelioglu, A., Cockburn, B., Qiu, W.: Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. Math. Comput. 86, 1643–1670 (2017).  https://doi.org/10.1090/mcom/3195 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47(2), 1092–1125 (2009).  https://doi.org/10.1137/080726653 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cockburn, B., Sayas, F.J.: Divergence-conforming HDG methods for Stokes flows. Math. Comput. 83, 1571–1598 (2014).  https://doi.org/10.1090/S0025-5718-2014-02802-0 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.J.: Analysis of HDG methods for Stokes flow. Math. Comput. 80(274), 723–760 (2011).  https://doi.org/10.1090/S0025-5718-2010-02410-X MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dobrev, V.A., Kolev, T.V., et al.: MFEM: Modular finite element methods. http://mfem.org (2018)
  11. 11.
    Hansbo, P., Larson, M.G.: Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191, 1895–1908 (2002).  https://doi.org/10.1016/S0045-7825(01)00358-9 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Henson, V.E., Yang, U.M.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41(1), 155–177 (2002).  https://doi.org/10.1016/S0168-9274(01)00115-5 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Howell, J.S., Walkington, N.J.: Inf-sup conditions for twofold saddle point problems. Numer. Math. 118, 663–693 (2011).  https://doi.org/10.1007/s00211-011-0372-5 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017).  https://doi.org/10.1137/15M1047696 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Labeur, R.J., Wells, G.N.: Energy stable and momentum conserving hybrid finite element method for the incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 34(2), A889–A913 (2012).  https://doi.org/10.1137/100818583 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lee, C.H., Vassilevski, P.S.: Parallel solver for \(H\)(div) problems using hybridization and AMG. In: Domain Decomposition Methods in Science and Engineering XXIII, pp. 69–80. Springer (2017).  https://doi.org/10.1007/978-3-319-52389-7_6 Google Scholar
  17. 17.
    Lehrenfeld, C., Schöberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Methods Appl. Mech. Eng. 307, 339–361 (2016).  https://doi.org/10.1016/j.cma.2016.04.025 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous galerkin method for stokes flow. Comput. Methods Appl. Mech. Eng. 199(9–12), 582–597 (2010).  https://doi.org/10.1016/j.cma.2009.10.007 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. J. Comput. Phys. 230(4), 1147–1170 (2011).  https://doi.org/10.1016/j.jcp.2010.10.032 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pestana, J., Wathen, A.J.: Natural preconditioning and iterative methods for saddle point systems. SIAM Rev. 57(1), 71–91 (2015).  https://doi.org/10.1137/130934921 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. 36(4), 1943–1967 (2016).  https://doi.org/10.1093/imanum/drv067 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rhebergen, S., Cockburn, B.: A space-time hybridizable discontinuous galerkin method for incompressible flows on deforming domains. J. Comput. Phys. 231(11), 4185–4204 (2012).  https://doi.org/10.1016/j.jcp.2012.02.011 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rhebergen, S., Wells, G.N.: Analysis of a hybridized/interface stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 55(4), 1982–2003 (2017).  https://doi.org/10.1137/16M1083839 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Rhebergen, S., Wells, G.N.: A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field. J. Sci. Comput. (2018).  https://doi.org/10.1007/s10915-018-0671-4 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wells, G.N.: Analysis of an interface stabilized finite element method: the advection-diffusion-reaction equation. SIAM J. Numer. Anal. 49(1), 87–109 (2011).  https://doi.org/10.1137/090775464 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK

Personalised recommendations