Skip to main content
SpringerLink
Log in

Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier–Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier–Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter \({\varepsilon}\), which means that the solution of the regularized problem converges to the solution of the Navier–Stokes type variational inequality problem as the parameter \({\varepsilon\longrightarrow 0}\). Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.

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. Glowinski R., Lions J.L., Trémolières R.: Numerical Analysis of Variational Inequalities. North Holland, Amsterdam (1981)

    MATH  Google Scholar 

  2. Glowinski R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York Inc. (1984)

    MATH  Google Scholar 

  3. Brezzi F., Hager W., Raviart P.A.: Error estimates for the finite element solution of variational inequalities—part 1: primal theory. Numer. Math. 28(4), 431–443 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brezzi F., Hager W., Raviart P.A.: Error estimates for the finite element solution of variational inequalities—part 2: mixed methods. Numer. Math. 31(1), 1–16 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  5. Falk R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28, 963–971 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  6. Scholz R.: Mixed finite element approximation of a fourth order variational inequality by the penalty method. Numer. Funct. Anal. Optim. 9(3–4), 233–247 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  7. Han W.M., Hua D.Y., Wang L.H.: Nonconforming finite element methods for a clamped plate with elastic unilateral obstacle. J. Integral Equ. Appl. 18(2), 267–284 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Shi D.Y., Chen S.C., Hagiwara I.: General error estimates of the nonconforming finite elements for a fourth order variational inequality with displacement obstacle. Math. Numer. Sin. 25, 99–106 (2003)

    MathSciNet  Google Scholar 

  9. Caboussat A., Glowinski R.: A two-grids/projection algorithm for obstacle problems. Comput. Math. Appl. 50, 171–178 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Caboussat A., Glowinski R.: A numerical method for a non-smooth advection-diffusion problem arising in sand mechanics. Commun. Pure. Appl. Anal. 8(1), 161–178 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. An R., Li K.T.: Stabilized mixed finite element approximation for fourth order obstacle problem (in Chinese). Acta Math. Appl. Sin. 32(6), 1068–1078 (2009)

    MATH  MathSciNet  Google Scholar 

  12. Wang L.H.: On the quadratic finite element approximation to the obstacle problem. Numer. Math. 92(4), 771–778 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dean E.J., Glowinski R.: Operator-splitting methods for the simulation of Bingham visco-plastic flow. Chin. Ann. Math. 23(2), 187–204 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dean E.J., Glowinski R., Guidoboni G.: On the numerical simulation of Bingham visco-plastic flow: old and new results. J. Non-Newtonian Fluid Mech. 142, 36–62 (2007)

    Article  MATH  Google Scholar 

  15. Latche J.C., Vola D.: Analysis of the Brezzi-Pitkaranta stabilized Galerkin scheme for creeping flows in Bingham fluids. SIAM J. Numer. Anal. 42(3), 1208–1225 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhang Y.M.: Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method. Numer. Math. 96(1), 153–184 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Han W.M., Wang L.H.: Nonconforming finite element analysis for a plate contact problem. SIAM J. Numer. Anal. 40(5), 1683–1697 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. An R., Li K.T.: Mixed finite element approximation for the plate contact problem (in Chinese). Acta Math. Sci. 30(3), 666–676 (2010)

    MathSciNet  Google Scholar 

  19. Ayadi, M., Gdoura, M.K., Sassi, T.: Error estimates for Stokes problem with Tresca friction condition. arXiv:1003.3352v1 (2010)

  20. Li Y., Li K.T.: Penalty finite element method for Stokes problem with nonlinear slip boundary conditions. Appl. Math. Comput. 204(1), 216–226 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li Y., Li K.T.: Pressure projection stabilized finite element method for Navier–Stokes equations with nonlinear slip boundary conditions. Computing 87, 113–133 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fujita, H.: Flow problems with unilateral boundary conditions. Lecons, Collège de France, October, 1993

  23. Fujita H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 888(1), 199–216 (1994)

    MATH  Google Scholar 

  24. Fujita H.: Non-stationary Stokes flows under leak boundary conditions of friction type. J. Comput. Math. 19, 1–8 (2001)

    MATH  MathSciNet  Google Scholar 

  25. Fujita H.: A coherent analysis of Stokes flows under boundary conditions of friction type. J. Comput. Appl. Math. 149(1), 57–69 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  26. Fujita H., Kawarada H.: Variational inequalities for the Stokes equation with boundary conditions of friction type. Recent development in domain decomposition methods and flow problems. GAKUTO Int. Ser. Math. Sci. Appl. 11, 15–33 (1998)

    MathSciNet  Google Scholar 

  27. Saito N., Fujita H.: Regularity of solutions to the Stokes equation under a certain nonlinear boundary condition. The Navier–Stokes equations. Lect. Notes Pure Appl. Math. 223, 73–86 (2001)

    MathSciNet  Google Scholar 

  28. Saito N.: On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions. Pub. RIMS Kyoto Univ. 40, 345–383 (2004)

    Article  MATH  Google Scholar 

  29. Li, Y.: Mathematical theories and numerical methods of the variational inequality problems in incompressible viscous fluid. Ph.D. Thesis, Xi’an Jiaotong University, Xi’an (2009)

  30. Heywood J.G., Rannacher R.: Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM Numer. Anal. 19(2), 275–311 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  31. Bochev P., Dohrmann C., Gunzburger M.: Stabilization of low-order mixed finite element. SIAM J. Numer. Anal. 44(1), 82–101 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Li J., He Y.: A stabilized finite element method based on two local Gauss integrations for the Stokes equations. J. Comput. Appl. Math. 214(1), 58–65 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Li J., He Y.N., Chen Z.: A new stabilized finite element method for the transient Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 197(1–4), 22–35 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Li J., He Y.N., Chen Z.: Performance of several stabilized finite element methods for the Stokes equations based on the lowest equal-order pairs. Computing 86(1), 37–51 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Li J.: Investigations on two kinds of two-level stabilized finite element methods for the stationary Navier–Stokes equations. Appl. Math. Comput. 182, 1470–1481 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Beirao da Veiga H.: Regularity for Stokes and generalized Stokes system under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equ. 9(9–10), 1079–1114 (2004)

    MATH  MathSciNet  Google Scholar 

  37. Beirao da Veiga H.: Regularity of solutions to a nonhomogeneous boundary value problem for general Stokes systems in \({R^n_+}\). Math. Ann. 331, 203–217 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Science, Wenzhou University, 325035, Wenzhou, China

    Yuan Li & Rong An

Authors
  1. Yuan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rong An
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong An.

Additional information

Supported by the National Natural Science Foundation of China (10901122 and 11001205).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, Y., An, R. Semi-discrete stabilized finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions based on regularization procedure. Numer. Math. 117, 1–36 (2011). https://doi.org/10.1007/s00211-010-0354-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-010-0354-z

Mathematics Subject Classification (2000)

Search

Navigation