A new quadrilateral MINI-element for Stokes equations

Oh-In Kwon; Chunjae Park

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 4, page 955-968
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kwon, Oh-In, and Park, Chunjae. "A new quadrilateral MINI-element for Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.4 (2014): 955-968. <http://eudml.org/doc/273124>.

@article{Kwon2014,
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.},
author = {Kwon, Oh-In, Park, Chunjae},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {MINI-element; superconvergence},
language = {eng},
number = {4},
pages = {955-968},
publisher = {EDP-Sciences},
title = {A new quadrilateral MINI-element for Stokes equations},
url = {http://eudml.org/doc/273124},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Kwon, Oh-In
AU - Park, Chunjae
TI - A new quadrilateral MINI-element for Stokes equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 4
SP - 955
EP - 968
AB - 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.
LA - eng
KW - MINI-element; superconvergence
UR - http://eudml.org/doc/273124
ER -

References

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  1. [1] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. CALCOLO21 (1984) 337–344. Zbl0593.76039MR799997
  2. [2] I. Babuška, The finite element method with Lagrange multipliers. Numer. Math.20 (1973) 179–192. Zbl0258.65108MR359352
  3. [3] W. Bai, The quadrilateral ‘Mini’ finite element for the Stokes problem. Comput. Methods Appl. Mech. Eng.143 (1997) 41–47. Zbl0895.76042MR1442388
  4. [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. Zbl0338.90047MR365287
  5. [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. Zbl0941.65115MR1726483
  6. [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. Zbl05879807MR2772093
  7. [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. Zbl1134.76026MR2311926
  8. [8] V. Girault and P.A. Raviart, Finite element methods for the Navier-Stokes equations: Theory and Algorithms. Springer-Verlag, New York (1986). Zbl0585.65077MR851383
  9. [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. Zbl1048.65114MR2004191
  10. [10] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Eq.8 (1992) 97–111. Zbl0742.76051MR1148797

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