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A new quadrilateral MINI-element for Stokes equations

Published online by Cambridge University Press:  30 June 2014

Oh-In Kwon  and
Chunjae Park
Oh-In Kwon
Affiliation:
Department of Mathematics, Konkuk University, 143-701 Seoul, Korea. cjpark@konkuk.ac.kr
Chunjae Park
Affiliation:
Department of Mathematics, Konkuk University, 143-701 Seoul, Korea. cjpark@konkuk.ac.kr
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Abstract

We introduce a new stable MINI-element pair for incompressible Stokes equations on quadrilateral meshes, which uses the smallest number of bubbles for the velocity. The pressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done with the standard Q1-conforming elements enriched by one bubble a quadrilateral. The superconvergence in the pressure of the proposed pair is analyzed on uniform rectangular meshes, and tested numerically on uniform and non-uniform meshes.

Keywords

MINI-elementsuperconvergence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 4 , July 2014 , pp. 955 - 968
Copyright
© EDP Sciences, SMAI, 2014

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References

Arnold, D.N., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. CALCOLO 21 (1984) 337–344. Google Scholar
Babuška, I., The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179–192. Google Scholar
Bai, W., The quadrilateral ‘Mini’ finite element for the Stokes problem. Comput. Methods Appl. Mech. Eng. 143 (1997) 41–47. Google Scholar
Brezzi, F., On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numer. R2 8 (1974) 129–151. Google Scholar
Douglas, J. Jr., Santos, J.E., Sheen, D. and Ye, X., Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. RAIRO: M2AN 33 (1999) 747–770. Google Scholar
Eichel, H., Tobiska, L. and Xie, H., Supercloseness and superconvergence of stabilized low order finite element discretization of the Stokes Problem. Math. Comput. 80 (2011) 697–722. Google Scholar
Franca, L.P., Oliveira, S.P. and Sarkis, M., Continuous Q1-Q1 Stokes elements stabilized with non-conforming null edge average velocity functions. Math. Models Meth. Appl. Sci. 17 (2007) 439–459. Google Scholar
V. Girault and P.A. Raviart, Finite element methods for the Navier-Stokes equations: Theory and Algorithms. Springer-Verlag, New York (1986).
Park, C. and Sheen, D., P 1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 624–640. Google Scholar
Rannacher, R. and Turek, S., Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Eq. 8 (1992) 97–111. Google Scholar