An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

This paper presents a nonconforming finite element scheme for the planar biharmonic equation which applis piecewise cubic polynomials ($P_3$) and possesses $\mathcal{O}(h^2)$ convergence rate in energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the $H^4$ functions with $\mathcal{O}(h^2)$ accuracy in broken $H^2$ norm. Besides, an equivalence $(\nabla_h^2\ \cdot,\nabla_h^2\ \cdot)=(\Delta_h\ \cdot,\Delta_h\ \cdot)$, which is usually not true for nonconforming finite element spaces, is proved on the newly designed spaces. The finite element space does not correspond to a finite element defined with Ciarlet's triple; however, a set of locally supported basis functions of the finite element space is still figured out. The notion of the finite element Stokes complex plays an important role in the analysis and also the construction of the basis functions.