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A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the 12 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 11. The nonconforming element consists ofP  3   ⊕  Span  {   x  3   y  −  x   y  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 andL  2   (  Ω  )    norms. Finally, numerical examples match our theoretical results very well. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 691–705, 2015

A new cubic nonconforming finite element on rectangles