Skip to main content
SpringerLink
Log in

Discontinuous Galerkin Methods for a Stationary Navier–Stokes Problem with a Nonlinear Slip Boundary Condition of Friction Type

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

Abstract

In this work, several discontinuous Galerkin (DG) methods are introduced and analyzed to solve a variational inequality from the stationary Navier–Stokes equations with a nonlinear slip boundary condition of friction type. Existence, uniqueness and stability of numerical solutions are shown for the DG methods. Error estimates are derived for the velocity in a broken \(H^1\)-norm and for the pressure in an \(L^2\)-norm, with the optimal convergence order when linear elements for the velocity and piecewise constants for the pressure are used. Numerical results are reported to demonstrate the theoretically predicted convergence orders, as well as the capability in capturing the discontinuity, the ability in handling the shear layers, the capacity in dealing with the advection-dominated problem, and the application to the general polygonal mesh of the DG methods.

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

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. An, R., Li, Y.: Two-level penalty finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions. Int. J. Numer. Anal. Model. 11, 608–623 (2014)

    MathSciNet  Google Scholar 

  3. Arnold, D.N.: An interior penalty finite element method with discontinous element. SIAM J. Numer. Anal. 19, 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. Bassi, F., Rebay, S., Mariotti, G., Pedinotti, S., Savini, M.: A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows. In: Decuypere, R., Dibelius, G. (eds.) proceedings of 2nd European Conference on Turbomachinery, Fluid Dynamics and thermodynamics, pp. 99–108. Technologisch Instituut, Antwerrpen (1997)

    Google Scholar 

  6. Brenner, S.: Korn’s inequalities for piecewise H\(^1\) vector fields. Math. Comput. 73, 1067–1087 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  8. Brezzi, F., Manzini, G., Marini, D., Pietra, P., Russo, A.: Discontinuous finite elements for diffusion problems. In: Atti Convegno in onore di F. Brioschi (Milan, 1997), Istituto Lombardo, Accademia di Scienze e Lettere, Milan, Italy, pp. 197–217 (1999)

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

    MATH  Google Scholar 

  10. Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local dicontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40, 319–343 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cockburn, B., Kanschat, G., Schotzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comput. 74, 1067–095 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Kanschat, G., Schotzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier–Stokes equations. J. Sci. Comput. 31, 61–73 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cockburn, B., Shu, C.-W.: The local dicontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Rev. Francaise Automat. Informat. Recherche Opérationnelle Sér. Rouge 7, 33–75 (1973)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity-part 1: formulation and analysis. Comput. Methods Appl. Mech. Eng. 196, 3881–3897 (2007)

    Article  MATH  Google Scholar 

  17. Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity-part 2: algorithms and numerical analysis. Comput. Methods Appl. Mech. Eng. 197, 1–21 (2007)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  19. Douglas Jr., J., Dupont, T.: Interior Penalty Procedures for Elliptic and Parabolic Galerkin Methods. Lecture Notes in Physics, vol. 58. Springer, Berlin (1976)

    Google Scholar 

  20. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

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

    Google Scholar 

  22. Fujita, H.: A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions. RIMS Kokyuroku 88, 199–216 (1994)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Fujita, H., Kawarada, H.: Variational inequalities for the Stokes equation with boundary conditions of friction type. Int. Ser. Math. Sci. Appl. 11, 15–33 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)

    MATH  Google Scholar 

  27. Girault, V., Riviere, B.: DG approxiamtion of coupled Navier–Stokes and Darcy equations by Beaver–Joseph–Saffman interface condition. SIAM J. Numer. Anal. 47, 2052–2089 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Girault, V., Riviere, B., Wheeler, M.F.: A discontinuous Galerkin method with non-overlapping domain decomposition for the Stokes and Navier–Stokes problems. Math. Comput. 74, 53–84 (2005)

    Article  MATH  Google Scholar 

  29. Girault, V., Riviere, B., Wheeler, M.F.: A splitting method using discontinuous Galerkin for the transient incompressible Navier–Stokes equations. M2AN Math. Model. Numer. Anal. 39, 1115–1147 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Girault, V., Scott, R.L.: A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40, 1–19 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Glowinski, R., Guidoboni, G., Pan, T.-W.: Wall-driven incompressible viscous flow in a two-dimensional semi-circular cavity. J. Comput. Phys. 40, 1–19 (2003)

    MATH  Google Scholar 

  32. Glowinski, R., Lions, J., Tremolieres, R.: Numerical Analysis of Variational Inequalities. Elsevier Science Ltd, North-Holland (1981)

    MATH  Google Scholar 

  33. Gudi, T., Porwal, K.: A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems. Math. Comput. 83, 257–278 (2014)

    MathSciNet  MATH  Google Scholar 

  34. Gudi, T., Porwal, K.: A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem. J. Comput. Appl. Math. 292, 257–278 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  35. Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity. AMS/IP Studies in Advanced Mathematics, vol. 30, 2nd edn. American Mathematical Society, Providence (2002)

    Book  MATH  Google Scholar 

  36. Jing, F.F., Li, J., Chen, Z.: Stabilized finite element methods for a blood flow model of arterosclerosis. Numer. Methods Partial Differ. Equ. 31, 2063–2079 (2015)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Kashiwabara, T.: On a strong solution of the non-stationary Navier–Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254, 756–778 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lazarov, R., Ye, X.: Stabilized discontinuous finite element approximation for Stokes equations. J. Comput. Appl. Math. 198, 236–252 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lesaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: de Boor, C.A. (ed.) Mathematical Aspects of Finite Element Methods in Partial Differential Equations. Academic Press, New York (1974)

    Google Scholar 

  41. Li, Y., An, R.: Two-level pressure projection finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions. Appl. Numer. Math. 61, 285–297 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. 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  MathSciNet  MATH  Google Scholar 

  43. Li, Y., Li, K.T.: Existence of the solution to stationary Navier–Stokes equations with nonlinear slip boundary conditions. J. Math. Anal. Appl. 381, 1–9 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, Y., Li, K.T.: 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 

  45. Liu, J.G., Shu, C.W.: A high-order discontinuous Galerkin method for 2D incompressible fows. J. Comput. Phys. 106, 577–596 (2000)

    Article  MATH  Google Scholar 

  46. Mu, L., Wang, J.P., Wang, Y.Q., Ye, X.: Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes. J. Comput. Appl. Math. 255, 432–440 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  47. Riviere, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  48. Riviere, B., Girault, V.: Discontinuous finite element methods for inompressible flows on subdomains with non-matching interfaces. Comput. Methods Appl. Mech. Eng. 195, 3274–3292 (2006)

    Article  MATH  Google Scholar 

  49. Riviere, B., Sardar, S.: Penalty-free discontinuous Galerkin methods for incompressible Navier–Stoke equation. Math. Model Methods Appl. Sci. 24, 1217–1236 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  50. Riviere, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part 1. Comput. Geosci. 3, 337–360 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. Saito, N., Fujita, H.: Regularity of solutions to the Stokes equation under a certain nonlinear boundary condition. Lect. Notes Pure Appl. Math. 223, 73–86 (2002)

    MathSciNet  MATH  Google Scholar 

  53. Schötzau, D., Schwab, C., Toselli, A.: Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal. 40, 2171–2194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  54. Temam, R.: Navier–Stokes Equations, Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (1983)

    MATH  Google Scholar 

  55. Wang, F., Han, W., Cheng, X.L.: Discontinuous Galerkin methods for solving elliptic variational inequalities. SIAM J. Numer. Anal. 48, 708–733 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  56. Wang, F., Han, W., Cheng, X.L.: Discontinuous Galerkin methods for solving the Signorini problem. IMA J. Numer. Anal. 31, 1754–1772 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  57. Wang, F., Han, W., Cheng, X.L.: Discontinuous Galerkin methods for solving a quasistatic contact problem. Numer. Math. 126, 771–800 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wang, F., Han, W., Eichholz, J., Cheng, X.L.: A posteriori error estimates for discontinuous Galerkin methods of obstacle problems. Nonlinear Anal. Real World Appl. 22, 664–679 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  60. Zeng, Y.P., Chen, J.R., Wang, F.: Error estimates of the weakly over-penalized symmetric interior penalty method for two variational inequalities. Comput. Math. Appl. 69, 760–770 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Feifei Jing

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Weimin Han, Wenjing Yan & Fei Wang

  3. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1410, USA

    Weimin Han

Authors
  1. Feifei Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weimin Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenjing Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wenjing Yan.

Additional information

This work is supported by the NSF under grant DMS-1521684 and by the NSF of China (Nos. 11371288, 11771350). The authors thank the referees for their valuable comments on the first version of the paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jing, F., Han, W., Yan, W. et al. 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). https://doi.org/10.1007/s10915-018-0644-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0644-7

Keywords

Search

Navigation