Skip to main content
SpringerLink
Log in

Decoupled two level finite element methods for the steady natural convection problem

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this work three kinds of decoupled two level finite element methods are proposed and analyzed for the natural convection problem. Firstly, some a priori bounds and the optimal error estimates of velocity and temperature in L 2 norm are provided for the standard Galerkin finite element method. Secondly, by using the coarse grid numerical solutions to decouple the nonlinear coupling term, we establish the convergence results for the proposed decoupled two level finite element schemes with meshes h and H satisfy h=H 2. Finally, two numerical examples are presented to show the efficiency and effectiveness of the proposed algorithms for the steady natural convection problem.

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. Boland, J., Layton, W.: Error analysis for finite element methods for steady natural convection problems. Numer. Fuct. Anal. And Optimiz. 11, 449–483 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991)

    MATH  Google Scholar 

  3. Cai, M.C., Mu, M.: A multilevel decoupled method for a mixed Stokes/Darcy model. J. Comput. Appl. Math. 236, 2452–2465 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Christon, M.A., Gresho, P.M., Sutton, S.B.: Computational predictability of time dependent natural convection flows in enclosures (including a benchmark solution). MIT Special Issue on Thermal Convection. Int. J. Numer. Methods Fluids 40, 953–980 (2002)

    Article  Google Scholar 

  5. Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)

    Book  MATH  Google Scholar 

  6. Cibik, A., Kaya, S.: A projection-based stabilized finite element method for steady-state natural convection problem. J. Math. Anal. Appl. 381, 469–484 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Damanik, H., Hron, J., Ouazzi, A., Turek, S.: A monolithic FEM-multigrid solver for non- isothermal incompressible flow on general meshes. J. Comput. Phys. 228, 3869–3881 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dibenedetto, E., Friedman, A.: Natural convection problems with change of phase. J. Differ. Equ. 62, 129–185 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  9. Elman, H.C., Mihajlovic, M.D., Silvester, D.J.: Fast iterative solvers for buoyancy driven flow problems. J. Comput. Phys. 230, 3900–3914 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ervin, V., Layton, W., Maubach, J.: A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations. Numer. Meth. Partial Differ. Equ. 12, 333–346 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gresho, P.M., Sani, R.L.: Incompressible Flow and the Finite Element Method, Isothermal Laminar Flow, Vol. 2. Wiley, Chichester (2000)

    Google Scholar 

  12. de Vahl Davis, D.: Natural convection of air in a square cavity: A benchmark solution. Internat. J. Numer. Methods Fluids 3, 249–264 (1983)

    Article  MATH  Google Scholar 

  13. He, Y.N.: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. He, Y.N.: Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with H 2 or H 1 initial data. Numer. Meth. Partial Differ. Equ. 21, 875–904 (2005)

    Article  MATH  Google Scholar 

  15. Hecht, F., Pironneau, O., Hyaric, A., Ohtsuka, K.: http://www.freefem.org/ff (2008)

  16. 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  MATH  MathSciNet  Google Scholar 

  17. John, V.: Residual a posteriori error estimates for two-level finite element methods for the Navier-Stokes equations, Appl. Numer. Math. 37, 503–518 (2001)

    Article  MATH  Google Scholar 

  18. Layton, W.J., Tran, H., Trenchea, C.: Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater-surface water flows. SIAM J. Numer. Anal. 51, 248–272 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Luo, Z.D.: The bases and applications of mixed finite element methods. Chinese Science Press, Beijing (2006)

    Google Scholar 

  20. Luo, Z.D., Lu, X.M.: A least squares Galerkin/Petrov mixed finite element method for the stationary natural convection. C. J. Numer. Math. Appl. 26, 17–37 (2004)

    Article  MathSciNet  Google Scholar 

  21. Massarotti, N., Nithiarasu, P., Zienkiewicz, O.C.: Characteristic-Based-Split (CBS) algorithm for incompressible flow problems with hear transfer. Internat. J. Numer. Methods Heat Fluid Flow 8, 969–990 (1998)

    Article  MATH  Google Scholar 

  22. Manzari, M.T.: An explicit finite element algorithm for convective heat transfer problems. Internat J. Numer. Methods Heat Fluid Flow 9, 860–877 (1999)

    Article  MATH  Google Scholar 

  23. Mu, M., Xu, J.C.: A two-grid method of a mixed stokes-darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45, 1801–1813 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mu, M., Zhu, X.H.: Decoupled schemes for a non-stationary mixed Stokes-Darcy model. Math. Comput. 79, 707–731 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Shen, S.M.: The finite element analysis of natural convection problem. J. Comput. Math. 16, 170–182 (1994). (Chinese)

    MATH  Google Scholar 

  26. Smethurst, C.A., Silvester, D.J., Mihajlovic, M.D.: Unstructured finite element method for the solution of the Boussinesq problem in three dimensions. Int. J. Numer. Methods Fluids 73, 791–812 (2013)

    MathSciNet  Google Scholar 

  27. Temam, R., 3rd: Navier-Stokes equation, Theory and numerical analysis. North-Holland, Amsterdam (1984)

    Google Scholar 

  28. Thompson, M.C., Ferziger, J.H.: An adaptive multigrid technique for the incompressible Navier-Stokes equations. J. Comput. Phys. 82, 94–121 (1989)

    Article  MATH  Google Scholar 

  29. Wan, D.C., Patnaik, B.S.V., Wei, G.W.: A new benchmark quality solution for the buoyancy driven cavity by discrete singular convolution. Numer. Heat Trans. Part B 40, 199–228 (2001)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Zhang, T., Yuan, J.Y.: Two novel decoupling algorithms for the steady Stokes-Darcy model based on two grid method. Dis. Conti. Dyna. Sys.-B 19, 849–865 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhang, T., Wu, Y.J. Stability and convergence of some decoupled schemes for the nonstationary Stokes-Darcy model, Submitted.

Download references

Author information

Authors and Affiliations

  1. School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 454003, People’s Republic China

    Tong Zhang

  2. Department of Geographical Science and Environment Engineering, Baoji University of Arts and Sciences, Baoji, 721013, People’s Republic China

    Xin Zhao

  3. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic China

    Pengzhan Huang

Authors
  1. Tong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xin Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pengzhan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tong Zhang.

Additional information

This work was supported by the NSF of China (No. 11301157, 11371031) and the Natural Science Foundation of Education Department of Henan Province (No.14A110008) and the Doctor Fund of Henan Polytechnic Univeristy (B2012-098) and the NCET-11-1041.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, T., Zhao, X. & Huang, P. Decoupled two level finite element methods for the steady natural convection problem. Numer Algor 68, 837–866 (2015). https://doi.org/10.1007/s11075-014-9874-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-014-9874-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation