A new cubic nonconforming finite element on rectangles

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.