Summary
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 orderN 2, whereN is the dimension of the corresponding finite element space. Hierarchical bases are introduced which give a condition number of orderN log(N)3.
Similar content being viewed by others
References
[AB] Axelsson, O., Barker, V.A.: Finite element solutions of boundary value problems, 1. Ed. London: Academic Press 1984
[BGP] Bristeau, M.O., Glowinski, R., Periaux, J.: Numerical methods for the Navier-Stokes equations. Comput. Phys. Report6, 73–188 (1987)
[CIA] Ciarlet, P.G.: The finite element method for elliptic problems, 1. Ed. New York: North Holland 1978
[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)
[D] Dörfler, W.: Hierarchical basis for elliptic problems. Universität Bonn: Thesis 1990
[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)
[GR] Griffiths, D.F.: Finite elements for incompressible flow. Math. Methods Appl. Sci.1, 16–31 (1979)
[V] Verführt, R.: On the preconditioning of non-conforming solenoidal finite element approximations of the stokes equation. Universität Zürich: Manuskript August 1989
[Y] Yserentant, H.: On the multi-level-splitting of finite element spaces. Numer. Math.49, 379–412 (1986)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dörfler, W. The conditioning of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics. Numer. Math. 58, 203–214 (1990). https://doi.org/10.1007/BF01385619
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01385619