A class of nonparametric DSSY nonconforming quadrilateral elements^∗

¹ Department of Mathematics, Ajou University, Suwon 443–749, Korea.
yjeon@ajou.ac.kr
² Department of Mathematics, Seoul National University, Seoul 151–747, Korea.
lamyun96@snu.ac.kr
³ Department of Mathematics, and Interdisciplinary Program in Computational Science and Technology, Seoul National University, Seoul 151–747, Korea.
sheen@snu.ac.kr
⁴ Department of Mathematics, Seoul National University, Seoul 151–747, Korea.
sim4322@snu.ac.kr

Received: 7 January 2013
Revised: 6 May 2013

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.