Skip to main content
SpringerLink
Log in

Analysis of a Reduced-Order HDG Method for the Stokes Equations

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

Abstract

In this paper, we analyze a hybridized discontinuous Galerkin method with reduced stabilization for the Stokes equations. The reduced stabilization enables us to reduce the number of facet unknowns and improve the computational efficiency of the method. We provide optimal error estimates in an energy and \(L^2\) norms. It is shown that the reduced method with the lowest-order approximation is closely related to the nonconforming Crouzeix–Raviart finite element method. We also prove that the solution of the reduced method converges to the nonconforming Gauss-Legendre finite element solution as a stabilization parameter \(\tau \) tends to infinity and that the convergence rate is \(O(\tau ^{-1})\).

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

Fig. 1

Similar content being viewed by others

References

  1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baran, Á., Stoyan, G.: Gauss-Legendre elements: a stable, higher order non-conforming finite element family. Computing 79(1), 1–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Becker, R., Capatina, D., Joie, J.: Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations. Numer. Methods Partial Differ. Equ. 28(3), 1013–1041 (2012)

    Article  MathSciNet  Google Scholar 

  5. 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 

  6. Burman, E., Stamm, B.: Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(1), 508–533 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burman, E., Stamm, B.: Local discontinuous Galerkin method with reduced stabilization for diffusion equations. Commun. Comput. Phys. 5(2–4), 498–514 (2009)

    MathSciNet  Google Scholar 

  8. Carrero, J., Cockburn, B., Schötzau, D.: Hybridized globally divergence-free LDG methods. I. The Stokes problem. Math. Comput. 75(254), 533–563 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cockburn, B., Cui, J.: An analysis of HDG methods for the vorticity–velocity–pressure formulation of the Stokes problem in three dimensions. Math. Comput. 81, 1355–1368 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cockburn, B., Cui, J.: Divergence-free HDG methods for the vorticity–velocity formulation of the Stokes problem. J. Sci. Comput. 52, 256–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47, 1092–1125 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cockburn, B., Nguyen, N.C., Peraire, J.: A comparison of HDG methods for Stokes flow. J. Sci. Comput. 45(1–3), 215–237 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cockburn, B., Sayas, F.J.: Divergence-conforming HDG methods for Stokes flows. Math. Comput. 83(288), 1571–1598 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cockburn, B., Shi, K.: Devising HDG methods for Stokes flow: an overview. Comput. Fluids 98, 221–229 (2014)

    Article  MathSciNet  Google Scholar 

  16. Crouzeix, M., Falk, R.S.: Nonconforming finite elements for the Stokes problem. Math. Comput. 52(186), 437–456 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. ESAIM. Math. Model. Numer. Anal. 7, 33–75 (1973)

    MathSciNet  MATH  Google Scholar 

  18. Egger, H., Waluga, C.: \(hp\) analysis of a hybrid DG method for Stokes flow. IMA J. Numer. Anal. 33(2), 687–721 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fortin, M., Soulie, M.: A nonconforming piecewise quadratic finite element on triangles. Int. J. Numer. Methods Eng. 19(4), 505–520 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Güzey, S., Cockburn, B., Stolarski, H.K.: The embedded discontinuous Galerkin method: application to linear shell problems. Int. J. Numer. Methods Eng. 70(7), 757–790 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lehrenfeld, C.: Hybrid Discontinuous Galerkin Methods for Solving Incompressible Flow Problems. Ph.D. Thesis, RWTH Aachen University (2010)

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oikawa, I.: A hybridized discontinuous Galerkin method with reduced stabilization. J. Sci. Comput. (2014). doi:10.1007/s10915-014-9962-6

  24. Qiu, W., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses (2013, submitted). arXiv:1312.1407

  25. Qiu, W., Shi, K.: An HDG method for convection diffusion equation. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0024-5

  26. Stoyan, G., Baran, Á.: Crouzeix-Velte decompositions for higher-order finite elements. Comput. Math. Appl. 51(6–7), 967–986 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science and Engineering, Waseda University, 3-4-1, Ohkubo, Shinjuku, Tokyo, 169-8555, Japan

    Issei Oikawa

Authors
  1. Issei Oikawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Issei Oikawa.

Additional information

This work was supported by JSPS KAKENHI Numbers 24224004 and 26800089.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Oikawa, I. Analysis of a Reduced-Order HDG Method for the Stokes Equations. J Sci Comput 67, 475–492 (2016). https://doi.org/10.1007/s10915-015-0090-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0090-8

Keywords

Search

Navigation