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In this paper, a new mixed finite element scheme using element-wisestabilization is introduced for the biharmonic equation with variablecoefficient on Lipschitz polyhedral domains. The proposed scheme doesn'tinvolve any integration along mesh interfaces. The gradient of the solution isapproximated by $H({\rm div})$-conforming $BDM_{k+1}$ element or vector valuedLagrange element with order $k+1$, while the solution is approximated byLagrange element with order $k+2$ for any $k\geq 0$.This scheme can be easilyimplemented and produces positive definite linear system. We provide a newdiscrete $H^{2}$-norm stability, which is useful not only in analysis of thisscheme but also in $C^{0}$ interior penalty methods and DG methods. Optimalconvergences in both discrete $H^{2}$-norm and $L^{2}$-norm are derived. Thisscheme with its analysis is further generalized to the von K\'arm\'anequations. Finally, numerical results verifying the theoretical estimates ofthe proposed algorithms are also presented.

A mixed finite element scheme for biharmonic equation with variable coefficient and von K\'arm\'an equations