On chorin's projection method for the incompressible navier-stokes equations

  • Rolf Rannacher
Numerical Methods
Part of the Lecture Notes in Mathematics book series (LNM, volume 1530)


Pseudo-compressibility methods are frequently used in computational fluid dynamics in order to cope with the algebraic difficulties caused by the incompressibility constraint. A popular example is the pressure stabilization (Petrov-Galerkin) method of T.J.R. Hughes, et al., which can be applied to the stationary as well as to the nonstationary Navier-Stokes problem. Also the classical projection method of A.J. Chorin can be interpreted as a variant of this method. This observation sheds some new light on the approximation properties of the projection method, particularly for the pressure.


Projection Method Pressure Stabilization Incompressibility Constraint Pressure Error Nonlinear Convective Term 
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  1. 1.
    Blum, H.: Asymptotic error expansion and defect correction in the finite element method. Habilitationsschrift, Universität Heidelberg, 1991.Google Scholar
  2. 2.
    Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math.Comp. 22, 745–762 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chorin, A.J.: On the convergence of discrete approximations of the Navier-Stokes equations. Math.Comp. 23, 341–353 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Durán, R., Nochetto, R.H.: Pointwise accuracy of a stable Petrov-Galerkin approximation to the Stokes problem. SIAM J.Numer.Anal. 26, 1395–1406 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin-Heidelberg 1986.CrossRefzbMATHGoogle Scholar
  6. 6.
    Gresho, P.M.: On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory. Int.J.Numer.Meth.Fluids, 11, 621–659 (1990). Part 2: Implementation. Int.J.Numer.Meth.Fluids 11, 587–620 (1990).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Harig,J.: Eine robuste und effiziente Finite-Elemente-Methode zur Lösung der inkompressiblen 3-D-Navier-Stokes-Gleichungen auf Vektorrechnern. Dissertation, Universität Heidelberg, 1991.Google Scholar
  8. 8.
    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). II. Stability of solutions and error estimates uniform in time. SIAM J.Numer.Anal. 23, 750–777 (1986). III. Smoothing property and higher order estimates for spatial discretization. SIAM J.Numer.Anal. 25, 489–512 (1988). IV. Error analysis for second-order time discretization. SIAM J.Numer. Anal. 27, 353–384 (1990).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid mechanics: V. Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolation. Comp.Meth.Appl.Mech.Eng. 59, 85–99 (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rannacher,R.: On pseudo-compressibility methods for the incompressible Navier-Stokes equations. In preparation.Google Scholar
  11. 11.
    Shen,J.: On error estimates of projection methods for the Navier-Stokes equations: First order schemes. To appear in SIAM J.Numer.Anal. 1991.Google Scholar
  12. 12.
    Shen,J.: Hopf bifurcation of the unsteady regularized driven cavity flows. Preprint Dept. of Mathematics, Indiana University-Bloomington, to appear in J.Comp.Phys. 1991.Google Scholar
  13. 13.
    Shen,J.: On error estimates of higher order projection and penalty-projection methods for Navier-Stokes equations. Indiana University-Bloomington, to appear in J.Comp.Phys. 1991.Google Scholar
  14. 14.
    Temam, R.: Sur l'approximation de la solution des equations de Navier-Stokes par la méthode des pas fractionaires II. Arch.Rati.Mech.Anal. 33, 377–385 (1969).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Temam,R.: Remark on the pressure boundary condition for the projection method. Technical Note 1991, to appear in Theoretical and Computational Fluid Dynamics.Google Scholar
  16. 16.
    Verfürth, R.: On the stability of Petrov-Galerkin formulations of the Stokes equations. Preprint, Universität Zürich, 1990.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Rolf Rannacher
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

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