Skip to main content
SpringerLink
Log in

Stokes equations under Tresca friction boundary condition: a truncated approach

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

Abstract

A priori error analysis of the finite element approximation of Stokes equations under slip boundary condition of friction type has been centered on the interpolation error on the slip zone. In this work, we propose a novel approach based on the approximation of the tangential component of traction force by a truncated (cutoff) function. More precisely, we carry out (i) a complete analysis of the truncated formulation from the continuous to discrete level in two and three dimensions. In particular, we show linear convergence rate of the finite element solution by assuming standard regularity of the weak solution. This improves all previous results. (ii) The description of our solution strategy, (iii) a verification of the convergence properties with analytic solution and benchmark tests.

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.

Similar content being viewed by others

References

  1. Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. In: mathematical Fluid Mechanics and Modeling, RIMS kōkyūroko, 888, Kyoto University, Kyoto 199–216 (1994)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Haslinger, J., Makinen, R.A.E.: The parameter identification in the Stokes system with threshold slip boundary conditions. J. Appl. Math. Mech. 100(6), ??? (2020)

    MathSciNet  Google Scholar 

  5. Haslinger, J., Stebel, J., Sassi, T.: Shape optimization for Stokes problem with threshold slip. Appl. Math. 59(6), 631–652 (2014)

    Article  MathSciNet  Google Scholar 

  6. Haslinger, J., Kucera, R., Satek, V., Sassi, T.: Stokes system with solution-dependent threshold slip boundary conditions: Analysis, approximation and implementation. Math. Mech. Solids. 23(3), 294–307 (2018)

    Article  MathSciNet  Google Scholar 

  7. Jing, F.F., Han, W., Zhang, Y.C., Yan, W.J.: Analysis of an aposteriori error estimator for a variational inequality governed by Stokes equations. J. Comput. Appl. Math. 372(11), 2020 (2721)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Ayadi, M., Baffico, L., Gdoura, M.K., Sassi, T. : Error estimates for Stokes problem with tresca friction conditions. Esaim:, M2AN. 48, 1413–1429 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Qiu, H., Xue, C., Xue, L.: Low-order stabilized finite element methods for the unsteady Stokes/Navier-Stokes equations with friction boundary conditions. MMAS. 41(5), 2119–2139 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Li, J, Zheng, H, Qingsong, Z: A priori and a posteriori estimates of the stabilized finite element methods for the incompressible flow with slip boundary conditions arising in arteriosclerosis. Advances in Difference Equations, 374 (2019)

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

    Article  MathSciNet  Google Scholar 

  14. Haslinger, J., Kucera, R., Satek, V.: Stokes system with local Coulomb’s slip boundary conditions: Analysis of discretized models and implementation. Comput. Math. with Appl. 77(6), 1655–1667 (2019)

    Article  MathSciNet  Google Scholar 

  15. Kucera, R., Haslinger, J, Satek, V., Jarosova, M.: Efficient methods for solving the Stokes problem with slip boundary conditions. Math. Comput. Simulat. 145, 114–124 (2018)

    Article  MathSciNet  Google Scholar 

  16. Ayadi, M., Ayed, H., Baffico, L., Sassi, T.: Stokes Problem with Slip Boundary Conditions of Friction type: Error Analysis of a Four-Field Mixed Variational Formulation. J. Sci. Comput. 81, 312–341 (2019)

    Article  MathSciNet  Google Scholar 

  17. Leroux, C.: Steady Stokes flows with threshold slip boundary conditions. Math. Model Methods Appl. Sci. 15(8), 1141–1168 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Djoko, J.K., Koko, J., Kucera, R.: power law Stokes equations with threshold slip boundary conditions: numerical methods and implementation. MMAS. 42(5), 1488–1511 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Djoko, J.K., Koko, J.: GLS methods for Stokes equations under boundary condition of friction type: formulation-analysis-numerical schemes and simulations. submitted (2021)

  21. Glowinski, R., Lions, J.-L., Trémolières, R.: Numerical Analysis of Variational Inequalities Studies in Mathematics and Its Applications, 8, North-Holland, Amsterdam (1981)

  22. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer Series in Computational Physics. Springer-Verlag, Berlin Heidelberg (2008)

    Google Scholar 

  23. Atkinson, K., Han, W.: Theoretical Numerical Analysis: a Functional Analysis Framework Texts in Applied Mathematics. Second Edition, vol. 39. Springer, Berlin (2005)

    Book  Google Scholar 

  24. Djoko, J.K.: Discontinuous Galerkin Finite element discretization for steady Stokes flows with threshold slip boundary conditions. Quaest. Math. 36, 501–516 (2013)

    Article  MathSciNet  Google Scholar 

  25. Jing, F., Han, W., Jing, 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  Google Scholar 

  26. Chouly, F., Hild, P.: On convergence of the panalty method for unilateral contact problem. Appl. Numer. Math. 65, 27–40 (2013)

    Article  MathSciNet  Google Scholar 

  27. Chouly, F., Fabre, M., Hild, P., Mlika, R., Pousin, J., Renard, Y.: An overview of recent results on nitsche’s method for contact problems, lecture notes in computational science and engineering 121, geometrically unfitted finite element methods and applications, 93-141 Springer (2018)

  28. Dione, I.: Optimal error estimates of the unilateral contact problem in curved and smooth boundary domain by the penalty method. IMA J. Numer. Anal. 40(1), 729–763 (2020)

    Article  MathSciNet  Google Scholar 

  29. Dione, I.: Optimal convergence analysis of the unilateral contact problem with and without Tresca friction conditions by the penalty method. J. Math. Anal. Appl. 472(1), 266–284 (2019)

    Article  MathSciNet  Google Scholar 

  30. Brezis, H.: Functional Analysis Sobolev Spaces and partial differential equations Springer (2010)

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

  32. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications Springer series in computational mathematics Springer Verlag Berlin (2013)

  33. Girault, V., Hecht, F.: Numerical Methods for Grade-Two Fluid models: Finite-Element Discretizations and Algorithms In Handbook of Numerical Analysis, Numerical Methods for Non-Newtonian Fluids. eds, R. glowinski, J. Xu. vol. XVI. North Holland; Amsterdam, 5–207 (2011)

  34. Lions, J.L.: Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires, Dunod, Paris (1969)

  35. Ciarlet, P.: Finite Element Method for Elliptic Problems. North Holland, Amstedam (1978)

  36. Glowinski, R.: Finite element methods for incompressible viscous flow. In Handbook of Numerical Analysis, P.G Ciarlet and J.L Lions, eds, vol. IX, North Holland; Amsterdam, 3–1176 (2003)

  37. Koko, J.: A MATLAB mesh generator for the two-dimensional finite element method. Appl. Math. Comput. 250, 650–664 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Koko, J.: Fast MATLAB assembly of FEM matrices in 2D and 3D using cell array approach. Int. J. Model. Simul. Sci. Comput. 7(02), 1650010 (2016)

    Article  Google Scholar 

  39. Koko, J.: Efficient, MATLAB Codes for 2D/3D Stokes equation with the mini-element. Informatica. 30, 243–268 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the two referees for their constructive remarks and comments.

Author information

Authors and Affiliations

  1. African Peer Review Mechanism, Johannesburg, South Africa

    Jules K. Djoko

  2. LIMOS, Université Clermont-Auvergne – CNRS UMR 6158, BP 10448, 63000, Clermont-Ferrand, France

    Jonas Koko

  3. Analyse Numerique, Universite de Yaounde I, Yaounde, Cameroon

    Sognia Konlack

Authors
  1. Jules K. Djoko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jonas Koko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sognia Konlack
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jules K. Djoko.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by: Jon Wilkening

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

Djoko, J.K., Koko, J. & Konlack, S. Stokes equations under Tresca friction boundary condition: a truncated approach. Adv Comput Math 48, 22 (2022). https://doi.org/10.1007/s10444-022-09933-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-022-09933-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation