Skip to main content
SpringerLink
Log in

Mixed Finite Element Method for a Hemivariational Inequality of Stationary Navier–Stokes Equations

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

Abstract

In this paper, we develop and study the mixed finite element method for a hemivariational inequality of the stationary Navier–Stokes equations (NS hemivariational inequality). The NS hemivariational inequality models the motion of a viscous incompressible fluid in a bounded domain, subject to a nonsmooth and nonconvex slip boundary condition. The incompressibility contraint is treated through a mixed formulation. Solution existence and uniqueness are explored. The mixed finite element method is applied to solve the NS hemivariational inequality and error estimates are derived. Numerical results are reported on the use of the P1b/P1 pair, illustrating the optimal convergence order predicted by the error analysis.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Alnaes, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M., Wells, G.: The FEniCS project version 1.5. Arch. Numer. Softw. 3, 100 (2015)

    Google Scholar 

  2. Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo 21, 337–344 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barboteu, M., Bartosz, K., Han, W., Janiczko, T.: Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. SIAM J. Numer. Anal. 53, 527–550 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, New York (2013)

    Book  MATH  Google Scholar 

  5. Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications. Springer, New York (2007)

    Book  MATH  Google Scholar 

  6. Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247–262 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  8. Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic, Plenum Publishers, Boston (2003)

    Book  MATH  Google Scholar 

  9. Djoko, J.K.: A priori error analysis for Navier–Stokes equations with slip boundary conditions of friction type. J. Math. Fluid Mech. 21, 32 (2019). (no. 1, Paper No. 17)

    Article  MathSciNet  MATH  Google Scholar 

  10. Djoko, J.K., Koko, J.: Numerical methods for the Stokes and Navier–Stokes equations driven by threshold slip boundary conditions. Comput. Methods Appl. Mech. Eng. 305, 936–958 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fang, C., Czuprynski, K., Han, W., Cheng, X.L., Dai, X.: Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition. IMA J. Numer. Anal. 40, 2696–2716 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feng, F., Han, W., Huang, J.: Virtual element method for elliptic hemivariational inequalities. J. Sci. Comput. 81, 2388–2412 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fujita, H.: Flow Problems with Unilateral Boundary Conditions. College de France, Lecons (1993)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  16. Han, W., Migórski, S., Sofonea, M.: A class of variational-hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Han, W., Sofonea, M.: Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer. 28, 175–286 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Han, W., Sofonea, M., Barboteu, M.: Numerical analysis of elliptic hemivariational inequalities. SIAM J. Numer. Anal. 55, 640–663 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Han, W., Sofonea, M., Danan, D.: Numerical analysis of stationary variational-hemivariational inequalities. Numer. Math. 139, 563–592 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, W., Wang, C.: Numerical analysis of a parabolic hemivariational inequality for semipermeable media. J. Comput. Appl. Math. 389, 113326 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Haslinger, J., Kučera, R., Šátek, V.: Stokes system with local Coulomb’s slip boundary conditions: analysis of discretized models and implementation. Comput. Math. Appl. 77, 1655–1667 (2019)

  22. Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht (1999)

    Book  MATH  Google Scholar 

  23. Jing, F., Han, W., Yan, W., Wang, F.: Discontinuous Galerkin methods for a stationary Navier–Stokes problem with a nonlinear slip boundary condition of friction type. J. Sci. Comput. 76, 888–912 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jing, F., Han, W., Zhang, Y., Yan, W.: Analysis of an a posteriori error estimator for a variational inequality governed by the Stokes equations. J. Comput. Appl. Math. 372, 112721 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kashiwabara, T.: On a finite element approximation of the Stokes equations under a slip boundary condition of the friction type. Japan J. Ind. Appl. Math. 30, 227–261 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kashiwabara, T.: Finite element method for Stokes equations under leak boundary condition of friction type. SIAM J. Numer. Anal. 51, 2448–2469 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Le Roux, C.: Steady Stokes flows with threshold slip boundary conditions. Math. Models Methods Appl. Sci. 15, 1141–1168 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Le Roux, C., Tani, A.: Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions. Math. Methods Appl. Sci. 30, 595–624 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, J., Jing, F., Chen, Z., Lin, X.: A priori and a posteriori estimates of stabilized mixed finite volume methods for the incompressible flow arising in arteriosclerosis. J. Comput. Appl. Math. 363, 35–52 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Y., An, R.: Penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions. Int. J. Numer. Methods Fluids 69, 550–566 (2012)

    Article  MathSciNet  Google Scholar 

  32. Li, Y., An, R.: Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem. J. Comput. Appl. Math. 290, 656–669 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, Y., Li, K.: Locally stablized finite element method for Stokes problem with nonlinear slip boundary conditions. J. Comput. Math. 28, 826–836 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, Y., Li, K.: Uzawa iteration method for Stokes type variational inequality of the second kind. Acta Math. Appl. Sin. 17, 303–316 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ling, M., Wang, F., Han, W.: The nonconforming virtual element method for a stationary Stokes hemivariational inequality with slip boundary condition. J. Sci. Comput. 85, article number 56 (2020)

  36. Migórski, S..: A note on optimal control problem for a hemivariational inequality modeling fluid flow. Conference Publications 2013(special), 545–554 (2013). https://doi.org/10.3934/proc.2013.2013.545

    Article  MathSciNet  MATH  Google Scholar 

  37. Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier–Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  38. Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Springer, New York (2013)

    Book  MATH  Google Scholar 

  39. Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications. Kluwer Academic Publishers, Boston (1999)

    Book  MATH  Google Scholar 

  40. Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York (1995)

    MATH  Google Scholar 

  41. Panagiotopoulos, P.D.: Nonconvex energy functions, hemivariational inequalities and substationary principles. Acta Mech. 42, 160–183 (1983)

    Google Scholar 

  42. Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  43. Qiu, H., An, R., Mei, L., Xue, C.: Two-step algorithms for the stationary incompressible Navier–Stokes equations with friction boundary conditions. Appl. Numer. Math. 120, 97–114 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  44. Saito, N.: On the Stokes equation with the leak or slip boundary conditions of friction type: regularity of solutions. Publ. Res. Inst. Math. Sci. 40, 345–383 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. CRC Press, Boca Raton (2018)

    MATH  Google Scholar 

  46. Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1979)

    MATH  Google Scholar 

  47. Temam, R.: The Navier–Stokes Equations and Nonlinear Functional Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM (1983)

  48. Wang, F., Ling, M., Han, W., Jing, F.: Adaptive discontinuous Galerkin methods for solving an incompressible Stokes flow problem with slip boundary condition of frictional type. J. Comput. Appl. Math. 371, 112700 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  49. Xu, W., Huang, Z., Han, W., Chen, W., Wang, C.: Numerical analysis of history-dependent variational-hemivariational inequalities with applications in contact mechanics. J. Comput. Appl. Math. 351, 364–377 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zeidler, E.: Nonlinear Functional Analysis and its Applications, Vol. II/B: Nonlinear Monotone Operators. Springer, New York (1990)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Weimin Han

  2. Program in Applied Mathematical and Computational Sciences (AMCS), University of Iowa, Iowa City, IA, 52242, USA

    Kenneth Czuprynski

  3. School of Mathematics and Statistics and Xi’an Key Laboratory of Scientific Computation and Applied Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, China

    Feifei Jing

Authors
  1. Weimin Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenneth Czuprynski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Feifei Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feifei Jing.

Additional information

Publisher's Note

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

This work is supported by the National Natural Science Foundation of China (No. 12001413) and the Postdoctoral Science Foundation of China (No. 2021M692647).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, W., Czuprynski, K. & Jing, F. Mixed Finite Element Method for a Hemivariational Inequality of Stationary Navier–Stokes Equations. J Sci Comput 89, 8 (2021). https://doi.org/10.1007/s10915-021-01614-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01614-9

Keywords

Mathematics Subject Classification

Search

Navigation