Skip to main content
SpringerLink
Log in

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navier-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems (with same stiffness matrix but different right-hand side) on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the algorithm produces a numerical solution with the optimal asymptotic H 2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

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. F Brezzi, M Fortin, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.

    Book  MATH  Google Scholar 

  2. R Bank, D Rose, Analysis of a multilevel iterative method for nonlinear finite element equations. Math. Comp. (1982), 160:453–465.

    Article  MathSciNet  Google Scholar 

  3. S C Brenner, L E Scott, The Mathematical Theory of Finite Element Methods (2nd ed), Springer, New York, 2002.

    MATH  Google Scholar 

  4. M Cayco. Finite Element Methods for the Stream Function Formulation of the Navier-Stokes Equations. PhD thesis, CMU, Pittsburgh, PA., 1985.

    Google Scholar 

  5. M Cayco, R A Nicolaides, Analysis of nonconforming stream function and pressure finite element spaces for the Navier-Stokes equations, Comp. and Math. Appl. (1989), 8:745–760.

    Article  MathSciNet  Google Scholar 

  6. M Cayco, R A Nicolaides, Finite element technique for optimal pressure recovery from stream function formulation of viscous flows, Math. Comp. (1986), 46:371–377.

    Article  MATH  MathSciNet  Google Scholar 

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

  8. X X Dai, X L Cheng, A two-grid method based on Newton iteration for the Navier-Stokes equtions. J. Comp. App. Math. (2008), 220:566–573.

    Article  MATH  MathSciNet  Google Scholar 

  9. H C Elman, J S David and J W Andrew, Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics, Oxford University Press, New York, 2005.

    MATH  Google Scholar 

  10. F Fairag, Two-level finite element method for the stream function formulation of the Navier-Stokes equations. Computers Math. Applic. (1998), 36:117–127.

    Article  MATH  MathSciNet  Google Scholar 

  11. F Fairag, A two-level discretization method for the stream function form of the Navier-Stokes equations, PHD thesis, University Pittsburgh PA, 1998.

  12. F Fairag, Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method, SIAM J. Sci. Comput. (2002), 24:1919–1929.

    Article  MathSciNet  Google Scholar 

  13. V. Girault, J. L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. (2001) 58:25–57.

    MATH  MathSciNet  Google Scholar 

  14. V Girault, P A Raviart, Finite Element approximation of the Navier-Stokes equations, Springer, Berlin, 1979.

    Book  MATH  Google Scholar 

  15. V Girault, P A Raviart, Finite Element Methods for the Navier-Stokes Equations Theory and Algorithm, Springer-Verlag, Berlin, 1986.

    Google Scholar 

  16. Max D Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms. Academic Press, London, 1989.

    MATH  Google Scholar 

  17. W Hackbusch, A Reusken, Analysis of a damped nonlinear multilevel method, Numer. Math. (1989), 55:225–246.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. W Layton, H W J Lenferink, A multilevel mesh independence principle for the Navier-Stokes equations, SIAM J. Numer. Anal. (1996), 33:17–30.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. W Layton, X Ye, Two-level discretizations of the stream function form of the navier-stokes equations Numer. Funct. Ana. Opt. (1999), 20:909–916.

    Article  MATH  MathSciNet  Google Scholar 

  22. C Taylor, P Hood, A numerical solution of Navier-Stokes equations using the finite element method. Comput. & Fluids (1986), 1:73–100.

    Article  MathSciNet  Google Scholar 

  23. R Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.

    MATH  Google Scholar 

  24. J Xu, A novel two-grid method for semilimear equtions. SIAM J. Sci. Comput. (1994), 15: 231–237.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. X Ye. Two-level discretizations with backtracking of the stream function form of the Navier-Stokes equations. Appl. Math. Comp. (1999), 100:131–138.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Xin-ping Shao & Dan-fu Han

Authors
  1. Xin-ping Shao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dan-fu Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin-ping Shao.

Additional information

This work was supported by National Foundation of Natural Science under the Grant 11071216.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shao, Xp., Han, Df. A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations. Appl. Math. J. Chin. Univ. 26, 368–378 (2011). https://doi.org/10.1007/s11766-011-2698-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-011-2698-2

MR Subject Classification

Keywords

Search

Navigation