  A new nonconforming rectangle element with cubic convergence for the energy
norm is introduced. The degrees of freedom (DOFs) are defined by the twelve
values at the three Gauss points on each of the four edges. Due to the
existence of one linear relation among the above DOFs, it turns out the DOFs
are eleven. The nonconforming element consists of $P_2\oplus
\Span\{x^3y-xy^3\}$. We count the corresponding dimension for Dirichlet and
Neumann boundary value problems of second-order elliptic problems. We also
present the optimal error estimates in both broken energy and $L_2(\O)$ norms.
Finally, numerical examples match our theoretical results very well.
