  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.
