A Mixed Finite Element Scheme for Biharmonic Equation with Variable Coefficient and von Kármán Equations (Communications in Computational Physics, 2022)
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 alongmesh interfaces. The gradient of the solution is approximated by $H$(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 $k≥0.$ This schemecan be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysisof this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with itsanalysis is further generalized to the von Kármán equations. Finally, numerical resultsverifying the theoretical estimates of the proposed algorithms are also presented.