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This paper introduces a new family of mixed finite elements for solving amixed formulation of the biharmonic equations in two and three dimensions. Thesymmetric stress $\bm{\sigma}=-\nabla^{2}u$ is sought in the Sobolev space$H({\rm{div}}\bm{div},\Omega;\mathbb{S})$ simultaneously with the displacement$u$ in $L^{2}(\Omega)$. Stemming from the structure of$H(\bm{div},\Omega;\mathbb{S})$ conforming elements for the linear elasticityproblems proposed by J. Hu and S. Zhang, the$H({\rm{div}}\bm{div},\Omega;\mathbb{S})$ conforming finite element spaces areconstructed by imposing the normal continuity of $\bm{div}\bm{\sigma}$ on the$H(\bm{div},\Omega;\mathbb{S})$ conforming spaces of $P_{k}$ symmetric tensors.The inheritance makes the basis functions easy to compute. The discrete spacesfor $u$ are composed of the piecewise $P_{k-2}$ polynomials without requiringany continuity. Such mixed finite elements are inf-sup stable on bothtriangular and tetrahedral grids for $k\geq 3$, and the optimal order ofconvergence is achieved. Besides, the superconvergence and the postprocessingresults are displayed. Some numerical experiments are provided to demonstratethe theoretical analysis.

A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids