Advertisement

Journal of Scientific Computing

, Volume 27, Issue 1–3, pp 97–110 | Cite as

An Efficient Discretization of the Navier–Stokes Equations in an Axisymmetric Domain. Part 1: The Discrete Problem and its Numerical Analysis

  • Z. Belhachmi
  • C. Bernardi
  • S. Deparis
  • F. Hecht
Article

Abstract

Any solution of the Navier–Stokes equations in a three-dimensional axisymmetric domain admits a Fourier expansion with respect to the angular variable, and it can be noted that each Fourier coefficient satisfies a variational problem on the meridian domain, all problems being coupled due to the nonlinear convection term. We propose a discretization of these equations which combines Fourier truncation and finite element methods applied to each two-dimensional system. We perform the a priori and a posteriori analysis of this discretization.

Keywords

Navier–Stokes equations Fourier truncation finite element method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belhachmi, Z., Bernardi, C., and Deparis, S. Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. To appear in Numer. Math.Google Scholar
  2. 2.
    Belhachmi, Z., Bernardi, C., Deparis, S., and Hecht, F. (2006). A truncated Fourier/finite element discretization of the Stokes equations in an axisymmetric domain. To appear in Math. Models Meth. Appl. Sci.Google Scholar
  3. 3.
    Bernardi C., Dauge M., Maday Y., Azaï ez M. (1999). Spectral Methods for Axisymmetric Domains, Series in Applied Mathematics 3. Gauthier–Villars & North-Holland.Google Scholar
  4. 4.
    Brezzi F., Rappaz J., Raviart P.-A. (1980). Finite dimensional approximation of nonlinear problems, Part I: Branches of nonsingular solutions. Numer. Math. 36, 1–25CrossRefMathSciNetGoogle Scholar
  5. 5.
    Girault, V., and Raviart, P.-A. (1986). Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer-Verlag.Google Scholar
  6. 6.
    Pousin J., Rappaz, J. (1994). Consistency, stability, a priori and a posteriori errors for Petrov–Galerkin methods applied to nonlinear problems. Numer. Math. 69, 213–231CrossRefMathSciNetGoogle Scholar
  7. 7.
    Verfürth, R. (1996). A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley & Teubner.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Z. Belhachmi
    • 1
  • C. Bernardi
    • 2
  • S. Deparis
    • 3
  • F. Hecht
    • 2
  1. 1.L.M.A.M. (U.M.R. 7122)Université de MetzMetz Cedex 01France
  2. 2.Laboratoire Jacques-Louis LionsC.N.R.S. & Université Pierre et Marie CurieParis Cedex 05France
  3. 3.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations