A new quadrilateral MINI-element for Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, p. 955-968
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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.

DOI : https://doi.org/10.1051/m2an/2013129
Classification:  65N30,  74S05,  76M10
Keywords: MINI-element, superconvergence
@article{M2AN_2014__48_4_955_0,
     author = {Kwon, Oh-In and Park, Chunjae},
     title = {A new quadrilateral MINI-element for Stokes equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {4},
     year = {2014},
     pages = {955-968},
     doi = {10.1051/m2an/2013129},
     zbl = {1299.76140},
     mrnumber = {3264342},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_4_955_0}
}
Kwon, Oh-In; Park, Chunjae. A new quadrilateral MINI-element for Stokes equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 955-968. doi : 10.1051/m2an/2013129. http://www.numdam.org/item/M2AN_2014__48_4_955_0/

[1] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. CALCOLO 21 (1984) 337-344. | MR 799997 | Zbl 0593.76039

[2] I. Babuška, The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179-192. | MR 359352 | Zbl 0258.65108

[3] W. Bai, The quadrilateral ‘Mini' finite element for the Stokes problem. Comput. Methods Appl. Mech. Eng. 143 (1997) 41-47. | MR 1442388 | Zbl 0895.76042

[4] F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numer. R2 8 (1974) 129-151. | Numdam | MR 365287 | Zbl 0338.90047

[5] J. Douglas Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. RAIRO: M2AN 33 (1999) 747-770. | Numdam | MR 1726483 | Zbl 0941.65115

[6] H. Eichel, L. Tobiska and H. Xie, Supercloseness and superconvergence of stabilized low order finite element discretization of the Stokes Problem. Math. Comput. 80 (2011) 697-722. | MR 2772093 | Zbl pre05879807

[7] L.P. Franca, S.P. Oliveira and M. Sarkis, Continuous Q1-Q1 Stokes elements stabilized with non-conforming null edge average velocity functions. Math. Models Meth. Appl. Sci. 17 (2007) 439-459. | MR 2311926 | Zbl 1134.76026

[8] V. Girault and P.A. Raviart, Finite element methods for the Navier-Stokes equations: Theory and Algorithms. Springer-Verlag, New York (1986). | MR 851383 | Zbl 0585.65077

[9] C. Park and D. Sheen, P1-nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 624-640. | MR 2004191 | Zbl 1048.65114

[10] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Eq. 8 (1992) 97-111. | MR 1148797 | Zbl 0742.76051