A class of nonparametric DSSY nonconforming quadrilateral elements

Youngmok Jeon; Hyun NAM; Dongwoo Sheen; Kwangshin Shim

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 6, page 1783-1796
  • ISSN: 0764-583X

Abstract

top
A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Calcolo 37 (2000) 253–254.], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.

How to cite

top

Jeon, Youngmok, et al. "A class of nonparametric DSSY nonconforming quadrilateral elements." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.6 (2013): 1783-1796. <http://eudml.org/doc/273173>.

@article{Jeon2013,
abstract = {A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Calcolo 37 (2000) 253–254.], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.},
author = {Jeon, Youngmok, NAM, Hyun, Sheen, Dongwoo, Shim, Kwangshin},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {nonconforming; finite element; quadrilateral; nonconforming quadrilateral elements; numerical examples},
language = {eng},
number = {6},
pages = {1783-1796},
publisher = {EDP-Sciences},
title = {A class of nonparametric DSSY nonconforming quadrilateral elements},
url = {http://eudml.org/doc/273173},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Jeon, Youngmok
AU - NAM, Hyun
AU - Sheen, Dongwoo
AU - Shim, Kwangshin
TI - A class of nonparametric DSSY nonconforming quadrilateral elements
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 6
SP - 1783
EP - 1796
AB - A new class of nonparametric nonconforming quadrilateral finite elements is introduced which has the midpoint continuity and the mean value continuity at the interfaces of elements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] The parametric DSSY element for general quadrilaterals requires five degrees of freedom to have an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Calcolo 37 (2000) 253–254.], while the new nonparametric DSSY elements require only four degrees of freedom. The design of new elements is based on the decomposition of a bilinear transform into a simple bilinear map followed by a suitable affine map. Numerical results are presented to compare the new elements with the parametric DSSY element.
LA - eng
KW - nonconforming; finite element; quadrilateral; nonconforming quadrilateral elements; numerical examples
UR - http://eudml.org/doc/273173
ER -

References

top
  1. [1] D.N. Arnold, D. Boffi and R.S. Falk, Approximation by quadrilateral finite elements. Math. Comput.71 (2002) 909–922. Zbl0993.65125MR1898739
  2. [2] S.C. Brenner and L.Y. Sung, Linear finite element methods for planar elasticity. Math. Comput.59 (1992) 321–338. Zbl0766.73060
  3. [3] Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: A correction. Calcolo37 (2000) 253–254. Zbl1012.65124MR1812789
  4. [4] Z. Cai, J. Douglas, Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo36 (1999) 215–232. Zbl0947.76047MR1740354
  5. [5] Z. Chen, Projection finite element methods for semiconductor device equations. Computers Math. Appl.25 (1993) 81–88. Zbl0772.65081MR1205422
  6. [6] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO – Math. Model. Numer. Anal.7 (1973) 33–75. Zbl0302.65087MR343661
  7. [7] J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems. ESAIM: M2AN 33 (1999) 747–770. Zbl0941.65115MR1726483
  8. [8] H. Han, Nonconforming elements in the mixed finite element method. J. Comput. Math.2 (1984) 223–233. Zbl0573.65083MR815417
  9. [9] J. Hu and Z.-C. Shi, Constrained quadrilateral nonconforming rotated Q1-element. J. Comput. Math.23 (2005) 561–586. Zbl1086.65111MR2190317
  10. [10] Y. Jeon, H. Nam, D. Sheen and K. Shim, A nonparametric DSSY nonconforming quadrilateral element with maximal inf-sup constant (2013). In preparation. Zbl1302.76104
  11. [11] M. Köster, A. Ouazzi, F. Schieweck, S. Turek and P. Zajac, New robust nonconforming finite elements of higher order. Appl. Numer. Math.62 (2012) 166–184. Zbl1238.65112MR2878019
  12. [12] R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg.124 (1995) 195–212. Zbl1067.74578MR1343077
  13. [13] C.-O. Lee, J. Lee and D. Sheen, A locking-free nonconforming finite element method for planar elasticity. Adv. Comput. Math.19 (2003) 277–291. Zbl1064.74165MR1973469
  14. [14] P. Ming and Z.-C. Shi, Nonconforming rotated Q1 element for Reissner-Mindlin plate. Math. Models Methods Appl. Sci.11 (2001) 1311–1342. Zbl1037.74048MR1859825
  15. [15] P. Ming and Z.-C. Shi, Two nonconforming quadrilateral elements for the Reissner–Mindlin plate. Math. Models Methods Appl. Sci.15 (2005) 1503–1517. Zbl1096.74050MR2168943
  16. [16] H. Nam, H.J. Choi, C. Park and D. Sheen, A cheapest nonconforming rectangular finite element for the Stokes problem. Comput. Methods Appl. Mech. Engrg.257 (2013) 77–86. Zbl1286.76044MR3043478
  17. [17] 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
  18. [18] C. Park and D. Sheen, A quadrilateral Morley element for biharmonic equations. Numer. Math.124 (2013) 395–413. Zbl1316.74063MR3054357
  19. [19] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ.8 (1992) 97–111. Zbl0742.76051MR1148797
  20. [20] Z.-C. Shi, An explicit analysis of Stummel’s patch test examples. Int. J. Numer. Meth. Engrg.20 (1984) 1233–1246. Zbl0557.65060MR751335
  21. [21] Z.C. Shi, The FEM test for convergence of nonconforming finite elements. Math. Comput.49 (1987) 391–405. Zbl0648.65075MR906178
  22. [22] S. Turek, Efficient solvers for incompressible flow problems, vol. 6. Lecture Notes in Comput. Sci. Engrg. Springer, Berlin (1999). Zbl0930.76002MR1691839
  23. [23] M. Wang, On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J. Numer. Anal.39 (2001) 363–384. Zbl1069.65116MR1860273
  24. [24] Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal.34 (1997) 640–663. Zbl0870.73074MR1442932

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.