A class of nonparametric DSSY nonconforming quadrilateral elements
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 6, pp. 1783-1796.

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.

DOI: 10.1051/m2an/2013088
Classification: 65N30
Keywords: nonconforming, finite element, quadrilateral
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     title = {A class of nonparametric {DSSY} nonconforming quadrilateral elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1783--1796},
     publisher = {EDP-Sciences},
     volume = {47},
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     year = {2013},
     doi = {10.1051/m2an/2013088},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2013088/}
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Jeon, Youngmok; NAM, Hyun; Sheen, Dongwoo; Shim, Kwangshin. A class of nonparametric DSSY nonconforming quadrilateral elements. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 6, pp. 1783-1796. doi : 10.1051/m2an/2013088. http://www.numdam.org/articles/10.1051/m2an/2013088/

[1] D.N. Arnold, D. Boffi and R.S. Falk, Approximation by quadrilateral finite elements. Math. Comput. 71 (2002) 909-922. | MR | Zbl

[2] S.C. Brenner and L.Y. Sung, Linear finite element methods for planar elasticity. Math. Comput. 59 (1992) 321-338. | Zbl

[3] Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen and X. Ye, Nonconforming quadrilateral finite elements: A correction. Calcolo 37 (2000) 253-254. | MR | Zbl

[4] Z. Cai, J. Douglas, Jr. and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215-232. | MR | Zbl

[5] Z. Chen, Projection finite element methods for semiconductor device equations. Computers Math. Appl. 25 (1993) 81-88. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

[8] H. Han, Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (1984) 223-233. | MR | Zbl

[9] J. Hu and Z.-C. Shi, Constrained quadrilateral nonconforming rotated Q1-element. J. Comput. Math. 23 (2005) 561-586. | MR | Zbl

[10] Y. Jeon, H. Nam, D. Sheen and K. Shim, A nonparametric DSSY nonconforming quadrilateral element with maximal inf-sup constant (2013). In preparation.

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[14] P. Ming and Z.-C. Shi, Nonconforming rotated Q1 element for Reissner-Mindlin plate. Math. Models Methods Appl. Sci. 11 (2001) 1311-1342. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[18] C. Park and D. Sheen, A quadrilateral Morley element for biharmonic equations. Numer. Math. 124 (2013) 395-413. | MR

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

[20] Z.-C. Shi, An explicit analysis of Stummel's patch test examples. Int. J. Numer. Meth. Engrg. 20 (1984) 1233-1246. | MR | Zbl

[21] Z.C. Shi, The FEM test for convergence of nonconforming finite elements. Math. Comput. 49 (1987) 391-405. | MR | Zbl

[22] S. Turek, Efficient solvers for incompressible flow problems, vol. 6. Lecture Notes in Comput. Sci. Engrg. Springer, Berlin (1999). | MR | Zbl

[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. | MR | Zbl

[24] Z. Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity. SIAM J. Numer. Anal. 34 (1997) 640-663. | MR | Zbl

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