  We introduce two pairs of stable cheapest nonconforming finite element space
pairs to approximate the Stokes equations. One pair has each component of its
velocity field to be approximated by the $P_1$ nonconforming quadrilateral
element while the pressure field is approximated by the piecewise constant
function with globally two-dimensional subspaces removed: one removed space is
due to the integral mean--zero property and the other space consists of global
checker--board patterns. The other pair consists of the velocity space as the
$P_1$ nonconforming quadrilateral element enriched by a globally
one--dimensional macro bubble function space based on $DSSY$
(Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure
field is approximated by the piecewise constant function with mean--zero space
eliminated. We show that two element pairs satisfy the discrete inf-sup
condition uniformly. And we investigate the relationship between them. Several
numerical examples are shown to confirm the efficiency and reliability of the
proposed methods.
