On the finite element approximation of the nonstationary Navier-Stokes problem

  • Rolf Rannacher
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 771)


In this note we report some basic convergence results for the semi-discrete finite element Galerkin approximation of the nonstationary Navier-Stokes problem. Asymptotic error estimates are established for a wide class of so-called conforming and nonconforming elements as described in the literature for modelling incompressible flows. Since the proofs are lengthy and very technical the present contribution concentrates on a precise statement of the results and only gives some of the key ideas of the argument for proving them. Complete proofs for the case of conforming finite elements may be found in a joint paper of J. Heywood, R. Rautmann and the author [5], whereas the nonconforming case will be treated in detail elsewhere.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland: Amsterdam 1978.zbMATHGoogle Scholar
  2. [2]
    Crouxeiz, M.; Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equation I. R.A.I.R.O. Anal. Numer. 3 (1973), 33–76.Google Scholar
  3. [3]
    Fortin, M.: Résolution des équations des fluides incompressibles par la méthode des éléments finis. In Proceedings of the Third International Conference on the Numerical Methods in Fluid Mechanics, Springer: Berlin-Heidelberg-New York 1972.Google Scholar
  4. [4]
    Heywood, J.G.: Classical solutions of the Navier-Stokes equations. In this Proceedings.Google Scholar
  5. [5]
    Heywood, J.G.; Rannacher, R.; Rautmann, R.: Semidiscrete finite element Galerkin approximation of the nonstationary Navier-Stokes problem. To appear.Google Scholar
  6. [6]
    Jamet, P.,; Raviart, P.-A.: Numerical solution of the stationary Navier-Stokes equation by finite element methods. In Computing Methods in Applied Sciences and Engineering, Part 1, Lecture Notes in Computer Sciences, Vol. 10, Springer: Berlin-Heidelberg-New York 1974.Google Scholar
  7. [7]
    Ladyžhenskaya, O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach: New York 1962.zbMATHGoogle Scholar
  8. [8]
    Rautmann, R.: On the convergence-rate of nonstationary Navier-Stokes approximations. In this Proceedings.Google Scholar
  9. [9]
    Temam, R.: Navier Stokes Equations. North-Holland: Amsterdam 1978.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Rolf Rannacher
    • 1
    • 2
  1. 1.Department of MathematicsThe University of MichiganAnn ArborUSA
  2. 2.Institut für Angewandte Mathematik der Universität BonnBonn 1Germany

Personalised recommendations