  A cheapest stable nonconforming finite element method is presented for
solving the incompressible flow in a square cavity without smoothing the corner
singularities. The stable cheapest nonconforming finite element pair based on
$P_1\times \widetilde{{\mathcal P}_{c}^h}$ on rectangular meshes
\cite{stab-cheapest} is employed with a minimal modification for the
discontinuous Dirichlet data on the top boundary, where $\widetilde{{\mathcal
P}_{c}^h}$ is the finite element space of piecewise constant pressures with the
globally one-dimensional checker-board pattern subspace eliminated. The
proposed Stokes elements have least number of degrees of freedom compared to
those of known stable Stokes elements. Three accuracy indications for our
elements are analyzed and numerically verified. Also, various numerous
computational results obtained by using our proposed element show excellent
accuracy.
