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.