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 $\mathbf{\sigma }=-{\nabla }^{2}u$ is sought in the Sobolev space $H(\mathrm{div}\mathbf{div},\Omega ;\mathbb{S})$ simultaneously with the displacement $u$ in $L^2(\Omega )$. Stemming from the structure of $H(\mathbf{div},\Omega ;\mathbb{S})$ conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the $H(\mathrm{div}\mathbf{div},\Omega ;\mathbb{S})$ conforming finite element spaces are constructed by imposing the normal continuity of $\mathbf{div}\mathbf{\sigma }$ on the $H(\mathbf{div},\Omega ;\mathbb{S})$ 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 $k\ge 3$, 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.

27 pages, 3 figures. Accept by SCI China Math

References in corpus (2)

• Complexes from complexes
• Finite elements for divdiv-conforming symmetric tensors