Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A stabilizer-free pressure-robust finite element method for the Stokes equations

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div) conforming finite element methods for the Stokes equations require stabilizers to enforce the weak continuity of velocity in tangential direction. Some stabilizers need to tune penalty parameter and some of them do not. Our method is stabilizer free although discontinuous velocity fields are used. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Extensive numerical investigations are conducted to test accuracy and robustness of the method and to confirm the theory. The numerical examples cover low- and high-order approximations up to the degree four, and 2D and 3D cases.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Al-Taweel, A., Wang, X.: A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method. Appl. Numer. Math. 150, 444–451 (2020)

    Article  MathSciNet  Google Scholar 

  2. Brenner, S., Scott, L.: The mathematical theory of finite element methods, 3rd edn. Springer, Berlin (2008)

    Book  Google Scholar 

  3. Brezzi, F., Fortin, M.: Mixed and hybrid finite elements. Springer-Verlag, New York (1991)

    Book  Google Scholar 

  4. Chen, W., Wang, F., Wang, Y.: Weak Galerkin method for the coupled Darcy–Stokes flow. IMA . Numer. Anal 36, 897–921 (2016)

    Article  MathSciNet  Google Scholar 

  5. Chen, G., Feng, M., Xie, X.: Robust globally divergence-free weak Galerkin methods for Stokes equations. J. Comput. Math. 34, 549–572 (2016)

    Article  MathSciNet  Google Scholar 

  6. Cockburn, B., Kanschat, G., Schotzau, D.: A note on discontinuous galerkin divergence-free solutions of the navier-stokes equations. J. Sci. Comp. 31, 61–73 (2007)

    Article  MathSciNet  Google Scholar 

  7. Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numer. 7, 33–76 (1973)

    MathSciNet  MATH  Google Scholar 

  8. Falk, R., Neilan, M.: Stokes complexes and the construction of stable finite element methods with pointwise mass conservation. SIAM J. Numer. Anal. 51, 1308–1326 (2013)

    Article  MathSciNet  Google Scholar 

  9. Girault, V., Raviart, P.: Finite element methods for the navier-stokes equations: theory and algorithms. Springer-Verlag, Berlin (1986)

    Book  Google Scholar 

  10. John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev.w 59, 492–544 (2017)

    Article  MathSciNet  Google Scholar 

  11. Lehrenfeld, C., Schoberl, J.: High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Comput. Method. Appl. Mech. Eng. 307, 339–361 (2016)

    Article  MathSciNet  Google Scholar 

  12. Li, R., Li, J., Liu, X., Chen, Z.: A weak Galerkin finite element method for a coupled Stokes-Darcy problem. Numer. Meth. Part. Diff. Eq. 33, 111–127 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations. Comput. Method. Appl. Mech. Eng 311, 304–326 (2016)

    Article  MathSciNet  Google Scholar 

  14. Mu, L., Wang, X., Ye, X.: A modified weak Galerkin finite element method for the Stokes equations. J. Comput. Appl. Math. 275, 79–90 (2015)

    Article  MathSciNet  Google Scholar 

  15. Schotzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171–2194 (2003)

    Article  MathSciNet  Google Scholar 

  16. Scott, L., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. ESAIM: M2AN 19, 111–143 (1985)

    Article  MathSciNet  Google Scholar 

  17. Tian, T., Zhai, Q., Zhang, R.: A new modified weak Galerkin finite element scheme for solving the stationary Stokes equations. J. Comput. Appl. Math. 329, 268–279 (2018)

    Article  MathSciNet  Google Scholar 

  18. Wang, J., Wang, Y., Ye, X.: A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods. SIAM J. Sci. Comput. 31, 2784–2802 (2009)

    Article  MathSciNet  Google Scholar 

  19. Wang, J., Ye, X.: New finite element methods in computational fluid dynamics by H(div) elements. SIAM Numer. Anal. 45, 1269–1286 (2007)

    Article  MathSciNet  Google Scholar 

  20. Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. Adv. Comput. Math. 42, 155–174 (2016)

    Article  MathSciNet  Google Scholar 

  21. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)

    Article  MathSciNet  Google Scholar 

  22. Wang, J., Ye, X.: A Weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comp. 83, 2101–2126 (2014)

    Article  MathSciNet  Google Scholar 

  23. Zhang, S.: Divergence-free finite elements on tetrahedral grids for k ≥ 6. Math. Comput. 80, 669–695 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Funding

Xiu Ye was supported in part by the National Science Foundation Grant DMS-1620016.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

Authors
  1. Xiu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shangyou Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shangyou Zhang.

Additional information

Communicated by: Lourenco Beirao da Veiga

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, X., Zhang, S. A stabilizer-free pressure-robust finite element method for the Stokes equations. Adv Comput Math 47, 28 (2021). https://doi.org/10.1007/s10444-021-09856-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-021-09856-9

Keywords

Mathematics Subject Classification 2010

Search

Navigation