Two $P_4$ Nonconforming Finite Elements for the Biharmonic Equation on Rectangular Meshes

In this paper, two new third order nonconforming finite element methods (NFEMs) on the rectangular grid are proposed for the biharmonic problem. Such finite elements are constructed by enriching 9 high-order polynomial bubbles to the $P_4$ polynomial space on each rectangle, so that the extra degrees of freedom can enforce the required sub-continuity of the finite element on the four edges. We prove that the methods are well-defined, have unique solution and converge at $\mathcal{O}(h^3)$ in the $H^2$ norm and $\mathcal{O}(h^5)$ in $L^2$ norm. The numerical results show that the new elements are very efficient.