Skip to main content
SpringerLink
Log in

Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper introduces a new stabilized finite element method for the coupled Stokes and Darcy problem based on the nonconforming Crouzeix-Raviart element. Optimal error estimates for the fluid velocity and pressure are derived. A numerical example is presented to verify the theoretical predictions.

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. Karper, T., Mardal, K. A., and Winther, R. Unified finite element discretizations of coupled Darcy-Stokes flow. Numerical Methods for Partial Differential Equation 25(2), 311–326 (2008)

    Article  MathSciNet  Google Scholar 

  2. Discacciati, M. and Quarteroni, A. Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations. Numerical Analysis and Advanced Applications, ENUMARH 2001 (eds. Brezzi, F. et al.), Springer, Milan, 3–20 (2001)

    Google Scholar 

  3. Discacciati, M. Domain Decomposition Method for the Coupling of Surface and Groundwater Flows, Ph. D. dissertation, École Polytechnique Fédérelede Lausanne, Switzerland (2004)

    Google Scholar 

  4. Arbogast, T. and Brunson, D. S. A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Computational Geosciences 11(3), 207–218 (2003)

    Article  MathSciNet  Google Scholar 

  5. Layton, W., Schieweck, F., and Yotov, I. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40(6), 2195–2218 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Rivière, B. Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problem. J. Sci. Comput. 22(1), 479–500 (2005)

    Article  MathSciNet  Google Scholar 

  7. Babuška, I. The finite element method with Lagrangian multiplier. Numer. Math. 20(1), 179–192 (1973)

    Article  MATH  Google Scholar 

  8. Brezzi, F. On the existence, uniqueness, and approximation of saddle point problems arsing from Lagrangian multipliers. RAIRO Anal. Numer. 2(8), 129–151 (1974)

    MathSciNet  Google Scholar 

  9. Rivi`ere, B. and Yotov, I. Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42(5), 1959–1977 (2005)

    Article  MathSciNet  Google Scholar 

  10. Urquiza, J. M., N’Dri, D., Garon, A., and Delfour, M. C. Coupling Stokes and Darcy equations. Appl. Numer. Math. 58(2), 525–538 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Burman, E. and Hansbo, P. Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem. Numerical Methods for Partial Diffential Equation 21(5), 986–997 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Crouzeix, M. and Raviart, P. A. Conforming and nonforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge 7(3), 33–75 (1973)

    MathSciNet  Google Scholar 

  13. Hansbo, P. and Larson, M. G. Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: Math. Model. Numer. Anal. 37(1), 63–72 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hansbo, P. and Larson, M. G. Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitshe’s method. Comput. Methods Appl. Mech. Engrg. 191(17–18), 1895–1908 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Beavers, G. S. and Joseph, D. D. Boundary conditions at a naturally impermeable wall. J. Fluid Mech. 30(2), 197–207 (2003)

    Google Scholar 

  16. Jäger, W. and Mikelić, A. On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(14), 1111–1127 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Saffman, P. On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50(1), 292–315 (1971)

    Google Scholar 

  18. Brezzi, F., Bristeau, M. O., Franca, K. L., Mallet, M., and Rogé, G. A relationship between stabilized finite element methods and the Galerkin method with bubble functions. Comput. Methods Appl. Mech. Engrg. 96(1), 117–129 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  19. Douglas, J. and Wang, J. An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495–508 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zhou, T. X. and Feng, M. F. A least squrares Petrov-Galerkin finite element method for the stationary Navier-Stokes equations. Math. Comput. 60(202), 531–543 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Thomée, V. Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin Heidelberg (1997)

    MATH  Google Scholar 

  22. Brenner, S. Korn’s inequalities for piecewise H 1 vector fields. Math. Comput. 73(247), 1067–1087 (2004)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, P. R. China

    Min-fu Feng  (冯民富), Rui-sheng Qi  (祁瑞生) & Rui Zhu  (朱瑞)

  2. Department of Information Technology, CNOOC Energy Technology and Services Limited Beijing Branch Company, Beijing, 100027, P. R. China

    Bing-tao Ju  (鞠炳焘)

Authors
  1. Min-fu Feng  (冯民富)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rui-sheng Qi  (祁瑞生)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rui Zhu  (朱瑞)
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bing-tao Ju  (鞠炳焘)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Min-fu Feng  (冯民富).

Additional information

Communicated by Zhe-wei ZHOU

Project supported by the Science and Technology Foundation of Sichuan Province (No. 05GG006-006-2)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Feng, Mf., Qi, Rs., Zhu, R. et al. Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem. Appl. Math. Mech.-Engl. Ed. 31, 393–404 (2010). https://doi.org/10.1007/s10483-010-0312-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-010-0312-z

Key words

Chinese Library Classification

2000 Mathematics Subject Classification

Search

Navigation