  In this paper, a new mixed finite element scheme using element-wise
stabilization is introduced for the biharmonic equation with variable
coefficient on Lipschitz polyhedral domains. The proposed scheme doesn't
involve any integration along mesh interfaces. The gradient of the solution is
approximated by $H({\rm div})$-conforming $BDM_{k+1}$ element or vector valued
Lagrange element with order $k+1$, while the solution is approximated by
Lagrange element with order $k+2$ for any $k\geq 0$.This scheme can be easily
implemented and produces positive definite linear system. We provide a new
discrete $H^{2}$-norm stability, which is useful not only in analysis of this
scheme but also in $C^{0}$ interior penalty methods and DG methods. Optimal
convergences in both discrete $H^{2}$-norm and $L^{2}$-norm are derived. This
scheme with its analysis is further generalized to the von K\'arm\'an
equations. Finally, numerical results verifying the theoretical estimates of
the proposed algorithms are also presented.
