Numerische Mathematik

, Volume 58, Issue 1, pp 203–214 | Cite as

The conditioning of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics

  • W. Dörfler


In order to solve the Stokes equations numerically, Crouzeix and Raviart introduced elements satisfying a discrete divergence condition. For the two dimensional case and uniform triangulations it is shown, that using the standard basis functions, the conditioning of the stiffness matrix is of orderN2, whereN is the dimension of the corresponding finite element space. Hierarchical bases are introduced which give a condition number of orderN log(N)3.

Subject classifications

AMS(MOS) 65F10 65N30 76D05 CR: G1.8 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AB] Axelsson, O., Barker, V.A.: Finite element solutions of boundary value problems, 1. Ed. London: Academic Press 1984Google Scholar
  2. [BGP] Bristeau, M.O., Glowinski, R., Periaux, J.: Numerical methods for the Navier-Stokes equations. Comput. Phys. Report6, 73–188 (1987)Google Scholar
  3. [CIA] Ciarlet, P.G.: The finite element method for elliptic problems, 1. Ed. New York: North Holland 1978Google Scholar
  4. [CR] Crouzeix, M., Raviart, P.A.: Conforming and nonconforming finite element methods for solving the stokes equations. R.A.I.R.O.7, 33–76 (1973)Google Scholar
  5. [D] Dörfler, W.: Hierarchical basis for elliptic problems. Universität Bonn: Thesis 1990Google Scholar
  6. [DT] Dobrowolski, M., Thomas, K.: Über die Struktur diskret divergenzfreier Finiter Elemente zur numerischen Approximation der Navier-Stokes-Gleichungen. Universität Bonn: SFB72 preprint nr. 679 (1984)Google Scholar
  7. [GR] Griffiths, D.F.: Finite elements for incompressible flow. Math. Methods Appl. Sci.1, 16–31 (1979)Google Scholar
  8. [V] Verführt, R.: On the preconditioning of non-conforming solenoidal finite element approximations of the stokes equation. Universität Zürich: Manuskript August 1989Google Scholar
  9. [Y] Yserentant, H.: On the multi-level-splitting of finite element spaces. Numer. Math.49, 379–412 (1986)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • W. Dörfler
    • 1
  1. 1.Institut für Angewandte MathematikBonn 1Germany

Personalised recommendations