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

A weak Galerkin finite element method for the stokes equations

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

Abstract

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k≥1 for the velocity and polynomials of degree k−1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k−1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.

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. 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 

  2. Babus̆ka, I.: The finite element method with Lagrangian multiplier. Numer.Math. 20, 179–192 (1973)

    Article  MathSciNet  Google Scholar 

  3. Beirao Da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brenner, S., Scott, R.: Mathematical theory of finite element methods. Springer (2002)

  5. Brezzi, F.: On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO, Anal. Numér. 2, 129–151 (1974)

    MathSciNet  Google Scholar 

  6. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Elements. Springer-Verlag, New York (1991)

    Book  Google Scholar 

  7. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland (1978)

  9. Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.: Analysis of HDG methods for Stokes flow. Math. Comput. 80(274), 723–760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. Girault, V., Raviart, P.A.: Finite Element Methods for the Navier-Stokes Equations: Theory and Algorithms. Springer-Verlag, Berlin (1986)

    Book  MATH  Google Scholar 

  12. Gunzburger, M.D.: Finite Element Methods for Viscous Incompressible Flows, A Guide to Theory, Practice and Algorithms. Academic, San Diego (1989)

    Google Scholar 

  13. Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on Polytopal Meshes. International J of Numerical Analysis and Modeling 12, 31–53 (2015). arXiv:1204.3655v2

    MathSciNet  Google Scholar 

  14. Mu, L., Wang, J., Ye, X.: A weak Galerkin finite element methods with polynomial reduction. J. Comp. and Appl. Math., in revision. arXiv:1304.6481

  15. Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comp. and Appl. Math. 241, 103–115 (2013). arXiv:1104.2897

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comp. 83(289), 2101–2126 (2014). S0025-5718(2014)02852-4. arXiv:1202.3655v2

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Junping Wang

  2. Division of Mathematical Sciences, National Science Foundation, Arlington, VA, 22230, USA

    Xiu Ye

Authors
  1. Junping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiu Ye.

Additional information

Communicated by: Jinchao Xu

This research was supported in part by National Science Foundation Grant DMS-1115097.

The research of Wang was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Ye, X. A weak Galerkin finite element method for the stokes equations. Adv Comput Math 42, 155–174 (2016). https://doi.org/10.1007/s10444-015-9415-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-015-9415-2

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation