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

Numerical Methods

Part of the Lecture Notes in Mathematics book series (LNM,volume 1530)

Abstract

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.

Keywords

  • Projection Method
  • Pressure Stabilization
  • Incompressibility Constraint
  • Pressure Error
  • Nonlinear Convective Term

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This work has been supported by the Deutsche Forschungsgemeinschaft, SFB 123, Universität Heidelberg. This paper is in final form and no similar paper has been or is being submitted elsewhere.

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References

  1. Blum, H.: Asymptotic error expansion and defect correction in the finite element method. Habilitationsschrift, Universität Heidelberg, 1991.

    Google Scholar 

  2. Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math.Comp. 22, 745–762 (1968).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Chorin, A.J.: On the convergence of discrete approximations of the Navier-Stokes equations. Math.Comp. 23, 341–353 (1969).

    CrossRef  MathSciNet  MATH  Google Scholar 

  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).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  5. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin-Heidelberg 1986.

    CrossRef  MATH  Google Scholar 

  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).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  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. 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).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  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).

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Rannacher,R.: On pseudo-compressibility methods for the incompressible Navier-Stokes equations. In preparation.

    Google Scholar 

  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. 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. 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. 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).

    MathSciNet  MATH  Google Scholar 

  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. Verfürth, R.: On the stability of Petrov-Galerkin formulations of the Stokes equations. Preprint, Universität Zürich, 1990.

    Google Scholar 

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© 1992 Springer-Verlag

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Rannacher, R. (1992). On chorin's projection method for the incompressible navier-stokes equations. In: Heywood, J.G., Masuda, K., Rautmann, R., Solonnikov, V.A. (eds) The Navier-Stokes Equations II — Theory and Numerical Methods. Lecture Notes in Mathematics, vol 1530. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090341

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  • DOI: https://doi.org/10.1007/BFb0090341

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