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

Abstract: This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric 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 elasticity problems proposed by J. Hu and S. Zhang, the $H({\rm{div}}\bm{div},\Omega;\mathbb{S})$ conforming finite element spaces are constructed 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 spaces for $u$ are composed of the piecewise $P_{k-2}$ polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for $k\geq 3$, and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis.