  In this paper, we consider a cubic $H^2$ nonconforming finite element scheme
$B_{h0}^3$ which does not correspond to a locally defined finite element with
Ciarlet$'$s triple but admit a set of local basis functions. For the first
time, we deduce and write out the expression of basis functions explicitly.
Distinguished from the most nonconforming finite element methods,
$(\delta\Delta_h\cdot,\Delta_h\cdot)$ with non-constant coefficient $\delta>0$
is coercive on the nonconforming $B_{h0}^3$ space which makes it robust for
numerical discretization. For fourth order eigenvalue problem, the $B_{h0}^3$
scheme can provide $\mathcal{O}(h^2)$ approximation for the eigenspace in
energy norm and $\mathcal{O}(h^4)$ approximation for the eigenvalues. We test
the $B_{h0}^3$ scheme on the vary-coefficient bi-Laplace source and eigenvalue
problem, further, transmission eigenvalue problem. Finally, numerical examples
are presented to demonstrate the effectiveness of the proposed scheme.
