Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Berlin (2013)

    MATH  Google Scholar 

  4. Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    Book  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  10. Dobrev, V.A., Kolev, T.V., et al.: MFEM: Modular finite element methods. http://mfem.org (2018)

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. 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

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, N2L 3G1, Canada

    Sander Rhebergen

  2. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB21PZ, UK

    Garth N. Wells

Authors
  1. Sander Rhebergen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Garth N. Wells
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sander Rhebergen.

Additional information

SR gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada through the Discovery Grant program (RGPIN-05606-2015) and the Discovery Accelerator Supplement (RGPAS-478018-2015).

Appendix A: Auxiliary Results

Appendix A: Auxiliary Results

We provide here some auxiliary results used in analyzing the preconditioners.

Defining \(a_h^{uu}(u_h, v_h) := a_h((u_h, 0), (v_h, 0))\), since \({|\!|\!| v_h |\!|\!|}_{DG} = {|\!|\!| (v_h, 0) |\!|\!|}_v\), a consequence of Eq. (12) is:

$$\begin{aligned} c_a^s{|\!|\!| v_h |\!|\!|}_{DG}^2 \le a_h^{uu}(v_h, v_h) \le c_a^b{|\!|\!| v_h |\!|\!|}_{DG}^2. \end{aligned}$$
(75)

Applying [11, Proposition 10] to a single cell K, the following inf-sup condition holds:

$$\begin{aligned} \beta _{DG}^K \parallel q_h \parallel _{K} \le \sup _{v_h \in V_h(K)} \frac{(q_h, \nabla \cdot v_h)_{K}}{{|\!|\!| v_h |\!|\!|}_{DG(K)}} \quad \forall q_{h} \in P_{k-1}(K), \end{aligned}$$
(76)

where \(\beta _{DG}^K > 0\) is a constant independent of h, \(V_h(K) := \mathinner { \left[ P_k(K)\right] } ^d\) and \({|\!|\!| v_h |\!|\!|}_{DG(K)}^2 := \parallel \nabla v_h \parallel ^2_{K} + \alpha h_K^{-1} \parallel v_h \parallel ^2_{\partial K}\). It follows that

$$\begin{aligned} \beta _{DG} \parallel q_h \parallel _{\Omega } \le \sup _{v_h \in V_h} \frac{(q_h, \nabla \cdot v_h)_{\mathcal {T}}}{{|\!|\!| v_h |\!|\!|}_{DG}}, \end{aligned}$$
(77)

where \(\beta _{DG} := \min _{K \in \mathcal {T}} \beta _{DG}^K\). Since \({|\!|\!| v_h |\!|\!|}_{DG} = {|\!|\!| (v_h, 0) |\!|\!|}_v\), it is easy to see from Eq. (26) that

$$\begin{aligned} \bar{\beta }_{DG} \parallel \bar{q}_h \parallel _{p} \le \sup _{v_h \in V_h} \frac{ \langle v_h \cdot n, \bar{q}_h \rangle _{\partial \mathcal {T}}}{{|\!|\!| v_h |\!|\!|}_{DG}}, \end{aligned}$$
(78)

where \(\bar{\beta }_{DG} > 0\) is a constant independent of h.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rhebergen, S., Wells, G.N. Preconditioning of a Hybridized Discontinuous Galerkin Finite Element Method for the Stokes Equations. J Sci Comput 77, 1936–1952 (2018). https://doi.org/10.1007/s10915-018-0760-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0760-4

Keywords

Search

Navigation