Skip to main content
Log in

Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary.

General error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature-dependent coefficients. These variable coefficients turn the diffusion and the buoyancy terms to be nonlinear, which increases the nonlinearity of the problems. An argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions. Error estimates are also provided for schemes modified by approximate coefficients, which are used conveniently in practical computations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. Bernardi, C., Métivet, B., and Pernaud-Thomas, B.: Couplage des équations de Navier–Stokes et de la chaleur: Le modèle et son approximation par éléments finis. RAIRO Modél. Math. Anal. Numér. 29, 871–921 (1995)

    Google Scholar 

  2. Boland, J., Layton, W.: An analysis of the finite element method for natural convection problems. Numer. Methods Partial Differential Equations 6, 115–126 (1990)

    Google Scholar 

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

    Google Scholar 

  4. Ciarlet, P.G.: The finite element method for elliptic problems. Classics in Applied Mathematics, Vol. 40, SIAM, 2002

  5. Clément, P.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9, 77–84 (1975)

    Google Scholar 

  6. Getling, A.V.: Rayleigh–Bénard convection: Structures and dynamics. World Scientific, 1998

  7. Girault, V., Raviart, P.-A.: Finite element approximation of the Navier–Stokes equations. Lecture Notes in Math., Vol. 749, Springer-Verlag, 1979

  8. Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations: Theory and algorithms. Springer-Verlag, 1986

  9. Guermond, J.-L., Quartapelle, L.: On the approximation of the unsteady Navier–Stokes equations by finite element projection methods. Numer. Math. 80, 207–238 (1998)

    Google Scholar 

  10. Heywood, J.G., 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)

    MATH  Google Scholar 

  11. Heywood, J.G., Rannacher, R.: Finite-element approximation of the nonstationary Navier–Stokes problem Part IV: Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27, 353–384 (1990)

    MATH  Google Scholar 

  12. Itoh, H., Tabata, M.: A finite element analysis for a thermal convection problem with the infinite Prandtl number. Hiroshima Math. J. 28, 555–570 (1998)

    Google Scholar 

  13. Lorca, S.A., Boldrini, J.L.: The initial value problem for a generalized Boussinesq model. Nonlinear Anal. 36, 457–480 (1999)

    Google Scholar 

  14. Ratcliff, J.T., Tackley, P.J., Schubert, G., Zebib, A.: Transitions in thermal convection with strongly variable viscosity. Phys. Earth Planet. Inter. 102, 201–212 (1997)

    Google Scholar 

  15. Tabata, M.: Finite element approximation to infinite Plandtl number Boussinesq equations with temperature-dependent coefficients—thermal convection problems in a spherical shell. submitted to Future Gener. Comput. Syst.

  16. Tabata, M.: Uniform solvability of finite element solutions in approximate domains. Japan J. Indust. Appl. Math. 18, 567–585 (2001)

    Google Scholar 

  17. Tabata, M., Suzuki, A.: A stabilized finite element method for the Rayleigh–Bénard equations with infinite Prandtl number in a spherical shell. Comput. Methods Appl. Mech. Engrg. 190 , 387–402 (2000)

    Google Scholar 

  18. Tabata, M., Suzuki, A.: Mathematical modeling and numerical simulation of Earth’s mantle convection. Mathematical Modeling and Numerical Simulation in Continuum Mechanics (Babuška, I., Ciarlet, P. G., and Miyoshi, T., eds.), Lect. Notes Comput. Sci. Eng., Vol. 19, Springer-Verlag, 2002, pp. 219–231

  19. Tabata, M., Tagami, D.: Error estimates for finite element approximations of drag and lift in nonstationary Navier–Stokes flows. Japan J. Indust. Appl. Math. 17, 371–389 (2000)

    Google Scholar 

  20. Tagami, D., Itoh, H.: A finite element analysis of thermal convection problems with the Joule heat. Japan J. Indust. Appl. Math. 20, 193–210 (2003)

    Google Scholar 

  21. Ungan, A., Viskanta, R.: Identification of the structure of the three dimensional thermal flow in an idling container glass melter. Glass Technology 28 , 252–260 (1987)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Masahisa Tabata.

Additional information

Mathematics Subject Classification (2000): 65M12, 65M60, 76M10

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tabata, M., Tagami, D. Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. 100, 351–372 (2005). https://doi.org/10.1007/s00211-005-0589-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-005-0589-2

Keywords

Navigation