Advertisement

Numerische Mathematik

, Volume 38, Issue 3, pp 309–332 | Cite as

On the transport-diffusion algorithm and its applications to the Navier-Stokes equations

  • O. Pironneau
Article

Summary

This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (ht+h×ht in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).

Subject Classifications

AMS(MOS): 65M25, 65N30 Cr: 5.17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bardos, C., Bercovier, M., Pironneau, O.: The Vortex Method with Finite Elements, Rapport de Recherche INRIA no 15 (1980)Google Scholar
  2. 2.
    Benque, J.P., Ibler, B., Keramsi, A., Labadie, G.: A Finite Element Method for Navier-Stokes Equations. Proceedings of the third International conference on finite elements in flow problems, Banff Alberta, Canada, 10–13 June, 1980Google Scholar
  3. 3.
    Bercovier, M., Pironneau, O.: Error Estimates for Finite Element Method Solution of the Stokes Problem in the Primitive variables. Numerische Mathematik,33, 211–224 (1979)Google Scholar
  4. 4.
    Bernadi, C.: Méthodes d'éléments finis mixtes pour les équations de Navier-Stokes, Thèse 3e cycle, Univ. Paris 6 (1979)Google Scholar
  5. 5.
    Boris, J.P., Book, D.L.: Flux corrected transport. SHASTA J. Comput. Phys.II, 36–69 (1973)Google Scholar
  6. 6.
    Crouzeix, M., Raviart, P.A.: Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO, R-3 (1973) pp. 33–76Google Scholar
  7. 7.
    Fortin, M., Thomasset, F.: Mixed finite element methods for incompressible flow problems. J. of comp. Physics31, 113–145 (1979)Google Scholar
  8. 8.
    Fritts, M.J., Boris, J.P.: The Lagrangian solution of transient problems in hydronamics using a triangular mesh. J. Comp. Ph.,31, 172–215 (1979)Google Scholar
  9. 9.
    Girault, V., Raviart, P.A.: Finite Element Approximation of Navier-Stokes equations. Lecture notes in Math. Berlin Göttingen Heidelberg: Springer 1979Google Scholar
  10. 10.
    Glowinski, R., Mantel, B., Periaux, J., Pironneau, O.: A Finite Element Approximation of Navier-Stokes Equations for Incompressible Viscous Fluids; Computer Methods in Fluids. Morgan Taylor Brebbia ed., Pentech Press 1980Google Scholar
  11. 11.
    Hecht, F.: Construction d'une base à divergence nulle pour un élément fini non conforme de degré 1 dansR 3 (to appear in RAIRO)Google Scholar
  12. 12.
    Heywood, J., Rannacker, R.: Finite Element Approximation of the nonstationary Navier-Stokes problem (to appear)Google Scholar
  13. 13.
    Lesaint, P., Raviart, P.A.: Résolution Numérique de l'Equation de Continuité par une Méthode du Type Elements Finis. Proc. Conference on Finite Elements in Rennes (1976)Google Scholar
  14. 14.
    Pironneau, O., Raviart, P.A., Sastri, V.: Finite Element Solution of the Transport Equations by convection plus projections (to appear)Google Scholar
  15. 15.
    Tabata, M.: A finite element approximation corresponding to the upwind differencing. Memoirs of Numerical Mathematics.1, 47–63 (1977)Google Scholar
  16. 16.
    Thomasset, F.: Finite Element Methods for Navier-Stokes Equations. VKI Lecture Series, March 25–29, 1980Google Scholar
  17. 17.
    Zienckiewicz, O., Heinrich, J.: The finite element method and convection problem in fluid mechanics. Finite Elements in Fluids (vol. 3), Gallagher ed., New York: Wiley 1978Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • O. Pironneau
    • 1
  1. 1.Département de MathématiquesCSP-Université de Paris-NordVilletaneuseFrance

Personalised recommendations