Skip to main content
SpringerLink
Log in

Two-Grid Arrow-Hurwicz Methods for the Steady Incompressible Navier-Stokes Equations

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

Abstract

In this paper, three kinds of two-grid Arrow-Hurwicz (A-H) methods are proposed and analyzed for the steady incompressible Navier-Stokes equations, which adopt the existing A-H method to obtain the coarse mesh solution, and further enhance the efficiency by three different one-step schemes (Oseen type, Simple type and Newton type) on the fine mesh. These methods combine the A-H method and the two-grid strategy, retaining the best features of two techniques and overcoming some of their limitations. Furthermore, the error analyses of the three methods are carefully studied and the numerical tests are reported to demonstrate the theoretical results and show the efficiency of the 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

Similar content being viewed by others

Availability of data and material

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

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

    MATH  Google Scholar 

  2. Arrow, K., Hurwicz, L., Uzawa, H.: Studies in Linear and Non-linear Programming. Stanford University Press, California (1958)

    MATH  Google Scholar 

  3. Ammi, A.A.O., Marion, M.: Nonlinear galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations. Numer. Math. 68, 189–213 (1994)

    Article  MathSciNet  Google Scholar 

  4. Boffi, D., Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, 2nd edn. Springer, Berlin (2013)

    MATH  Google Scholar 

  5. Bramale, J.H., Xu, J.: Some estimates for a weighted \(L^2\) projection. Math. Comp. 56, 463–476 (1991)

    MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  7. Bramble, J.H., Pasciak, J.E., Vassilev, A.T.: Uzawa type algorithms for nonsymmetric saddle point problems. Math. Comp. 69, 667–689 (2000)

    Article  MathSciNet  Google Scholar 

  8. Cao, Y., Dong, J., Wang, Y.: A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier-Stokes. J. Comput. Appl. Math. 273, 41–60 (2015)

    Article  MathSciNet  Google Scholar 

  9. Chen, L., Hu, X., Wang, M., Xu, J.: A multigrid solver based on distributive smoother and residual overweighting for Oseen problems. Numer. Math. Theor. Meth. Appl. 8, 237–252 (2015)

    Article  MathSciNet  Google Scholar 

  10. Chorin, A.J., Marsden, J.E.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Springer, New York (1993)

    Book  Google Scholar 

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

    MATH  Google Scholar 

  12. Chen, P., Huang, J., Sheng, H.: Solving steady incompressible Navier-Stokes equations by the Arrow-Hurwicz method. J. Comput. Appl. Math. 311, 100–114 (2017)

    Article  MathSciNet  Google Scholar 

  13. Du, B., Huang, J.: The generalized Arrow-Hurwicz method with applications to fluid computation. Commun. Comput. Phys. 25, 752–780 (2019)

    Article  MathSciNet  Google Scholar 

  14. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: with applications in incompressible fluid dynamics, 2nd edn. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  15. Franca, L.P., Hughes, T.J.R.: Convergence analyses of Galerkin least-squares methods for symmetric advectiveCdiffusive forms of the Stokes and incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 105, 285–298 (1993)

    Article  Google Scholar 

  16. Girault, V., Raviart, P.A.: Finite Element Approximations of the Navier Stokes Equations. Springer, New York (1986)

    Book  Google Scholar 

  17. Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)

    Article  Google Scholar 

  18. Girault, V., Lions, J.L.: Two-level finite element scheme for the transient Navier-Stokes problem. Math. Model. Numer. Anal. 35, 945–980 (2001)

    Article  Google Scholar 

  19. He, Y., Huang, A.: A simplified two-level method for the steady Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 197, 1568–1576 (2008)

    Article  MathSciNet  Google Scholar 

  20. He, Y., Li, K.: Two-level stabilized finite element methods for the steady Navier-Stokes problem. Computing 74, 337–351 (2005)

    Article  MathSciNet  Google Scholar 

  21. He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198, 1351–1359 (2009)

    Article  MathSciNet  Google Scholar 

  22. Huang, P., Feng, X., He, Y.: Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier-Stokes equations. Appl. Math. Model. 37, 728–741 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. John, V., Kaya, S.: A finite element variational multiscale method for the Navier-Stokes equations. SIAM J. Sci. Comput. 26, 1485–1503 (2005)

    Article  MathSciNet  Google Scholar 

  25. Layton, W.: A two-level discretization method for the Navier-Stokes equations. Comput. Math. Appl. 26, 33–38 (1993)

    Article  MathSciNet  Google Scholar 

  26. Layton, W., Tobiska, L.: A two-level method with backtracking for the Navier-Stokes equations. SIAM J. Numer. Anal. 35, 2035–2054 (1998)

    Article  MathSciNet  Google Scholar 

  27. Li, K., Hou, Y.: An AIM and one-step Newton method for the Navier-Stokes equations. Comput. Meth. Appl. Mech. Eng. 190, 6141–6155 (2001)

    Article  MathSciNet  Google Scholar 

  28. Nochetto, R.H., Pyo, J.H.: Optimal relaxation parameter for the Uzawa method. Numer. Math. 98, 695–702 (2004)

    Article  MathSciNet  Google Scholar 

  29. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    Book  Google Scholar 

  30. Sagaut, P.: Large Eddy Simulation for Incompressible Flows, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  31. Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis, Amsterdam (1984)

    MATH  Google Scholar 

  32. Wathen, A.J.: Preconditioning. Acta Numerica 24, 329–376 (2015)

    Article  MathSciNet  Google Scholar 

  33. Xu, H., He, Y.: Some iterative finite element methods for steady Navier-Stokes equations with different viscosities. J. Comput. Phys. 232, 136–152 (2013)

    Article  MathSciNet  Google Scholar 

  34. Xu, J.: A novel two-grid method for semi-linear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)

    Article  MathSciNet  Google Scholar 

  35. Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)

    Article  MathSciNet  Google Scholar 

  36. Zheng, H., Hou, Y., Shi, F., Song, L.: A finite element variational multiscale method for incompressible flows based on two local Gauss integrations. J. Comput. Phys. 228(16), 5961–5977 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank two anonymous reviewers for valuable comments which helped to improve an early version of the paper.

Funding

The work of Prof. Jianguo Huang was partially supported by the National Key Research and Development Project (2020YFA0709800) and NSFC (Grant No. 12071289). The work of Prof. Haibiao Zheng was partially supported by NSFC (Grant No. 11971174), NSF of Shanghai (Grant No. 19ZR1414300) and Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).

Author information

Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China

    Binbin Du

  2. School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Jianguo Huang

  3. School of Mathematical Sciences, East China Normal University, Shanghai, 200241, China

    Haibiao Zheng

Authors
  1. Binbin Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianguo Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haibiao Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this manuscript.

Corresponding author

Correspondence to Jianguo Huang.

Ethics declarations

Conflicts of interest/Competing interests

The authors have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this manuscript.

Code availability

The codes during the current study are available from the corresponding author on reasonable request.

Additional information

Publisher's Note

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

JH was partially supported by the National Key Research and Development Project (2020YFA0709800) and NSFC (Grant No. 12071289). HZ was partially supported by NSFC (Grant No. 11971174), NSF of Shanghai (Grant No. 19ZR1414300) and Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, B., Huang, J. & Zheng, H. Two-Grid Arrow-Hurwicz Methods for the Steady Incompressible Navier-Stokes Equations. J Sci Comput 89, 24 (2021). https://doi.org/10.1007/s10915-021-01627-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-021-01627-4

Keywords

Search

Navigation