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More pressure in the finite element discretization of the Stokes problem

Published online by Cambridge University Press:  15 April 2002

Christine Bernardi  and
Frédéric Hecht
Christine Bernardi
Affiliation:
Analyse Numérique, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. (bernardi@ann.jussieu.fr)
Frédéric Hecht
Affiliation:
Analyse Numérique, C.N.R.S. & Université Pierre et Marie Curie, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France. (bernardi@ann.jussieu.fr)
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Abstract

For the Stokes problem in a two- or three-dimensional bounded domain, we propose a new mixed finite element discretization which relies on a nonconforming approximation of the velocity and a more accurate approximation of the pressure. We prove that the velocity and pressure discrete spaces are compatible, in the sense that they satisfy an inf-sup condition of Babuška and Brezzi type, and we derive some error estimates.

Keywords

Finite elementsStokes probleminf-sup conditiondivergence-free basis functions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 5 , September 2000 , pp. 953 - 980
Copyright
© EDP Sciences, SMAI, 2000

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References

K. Arrow, L. Hurwicz and H. Uzawa, Studies in Nonlinear Programming. Stanford University Press, Stanford (1958).
Babuska, I., The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. CrossRef
C. Bergé, Théorie des graphes. Dunod, Paris (1970).
Boland, J. and Nicolaides, R., Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20 (1983) 722-731. CrossRef
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO - Anal. Numér. 8 R2 (1974) 129-151.
P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in the Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.-L. Lions Eds., North-Holland, Amsterdam (1991) 17-351.
P. Clément, Développement et applications de méthodes numériques volumes finis pour la description d'écoulements océaniques. Thesis, Université Joseph Fourier, Grenoble (1996).
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO - Anal. Numér. 7 R3 (1973) 33-76.
P. Emonot, Méthodes de volumes éléments finis: application aux équations de Navier-Stokes et résultats de convergence. Thesis, Université Claude Bernard, Lyon (1992).
M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO - Anal. Numér. 11 R3 (1977) 341-354.
V. Girault and P.-A. Raviart, Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin (1986).
F. Hecht, Construction d'une base d'un élément fini P 1 non conforme à divergence nulle dans $\mathbb{R}^3$ . Thesis, Université Pierre et Marie Curie, Paris (1980).
Hecht, F., Construction d'une base de fonctions P 1 non conforme à divergence nulle dans $\mathbb{R}^3$ . RAIRO - Anal. Numér. 15 (1981) 119-150. CrossRef
Verfürth, R., Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO - Anal. Numér. 18 (1984) 175-182. CrossRef