More pressure in the finite element discretization of the Stokes problem

Christine Bernardi; Frédéric Hecht

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 5, page 953-980
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Bernardi, Christine, and Hecht, Frédéric. "More pressure in the finite element discretization of the Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis 34.5 (2010): 953-980. <http://eudml.org/doc/197510>.

@article{Bernardi2010,
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. },
author = {Bernardi, Christine, Hecht, Frédéric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite elements; Stokes problem; inf-sup condition; divergence-free basis functions.; mixed finite element discretization; nonconforming approximation of velocity; approximation of pressure; error estimates},
language = {eng},
month = {3},
number = {5},
pages = {953-980},
publisher = {EDP Sciences},
title = {More pressure in the finite element discretization of the Stokes problem},
url = {http://eudml.org/doc/197510},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Bernardi, Christine
AU - Hecht, Frédéric
TI - More pressure in the finite element discretization of the Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 5
SP - 953
EP - 980
AB - 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.
LA - eng
KW - Finite elements; Stokes problem; inf-sup condition; divergence-free basis functions.; mixed finite element discretization; nonconforming approximation of velocity; approximation of pressure; error estimates
UR - http://eudml.org/doc/197510
ER -

References

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  2. I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math.20 (1973) 179-192.  Zbl0258.65108
  3. C. Bergé, Théorie des graphes. Dunod, Paris (1970).  
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  5. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO - Anal. Numér.8 R2 (1974) 129-151.  Zbl0338.90047
  6. 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.  Zbl0875.65086
  7. P. Clément, Développement et applications de méthodes numériques volumes finis pour la description d'écoulements océaniques. Thesis, Université Joseph Fourier, Grenoble (1996).  
  8. 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.  Zbl0302.65087
  9. P. Emonot, Méthodes de volumes éléments finis: application aux équations de Navier-Stokes et résultats de convergence. Thesis, Université Claude Bernard, Lyon (1992).  
  10. M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO - Anal. Numér.11 R3 (1977) 341-354.  
  11. V. Girault and P.-A. Raviart, Finite Element Methods for the Navier-Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin (1986).  Zbl0585.65077
  12. F. Hecht, Construction d'une base d'un élément fini P1 non conforme à divergence nulle dans 3 . Thesis, Université Pierre et Marie Curie, Paris (1980).  
  13. F. Hecht, Construction d'une base de fonctions P1 non conforme à divergence nulle dans 3 . RAIRO - Anal. Numér.15 (1981) 119-150.  Zbl0471.76028
  14. R. Verfürth, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO - Anal. Numér.18 (1984) 175-182.  Zbl0557.76037

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