论文标题
三角形和四面体网格上的双旋转方程的混合有限元元素
A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids
论文作者
论文摘要
本文介绍了一个新的混合有限元素系列，用于在两个维度和三个维度中求解Biharmonic方程的混合配方。在Sobolev space $ h（{{\ rm {div}} \ bm {div}，ω; \ mathbb {s}）$同时与位移$ u $ in $ l^{2}（2} $Ω同时，对称应力$ \bmσ= - \bmσ= - \bmσ= - \bmσ= - \bmσ= - \bmσ= - \bmσ= - \bmσ源于$ h（\ bm {div}，ω; \ mathbb {s}）$符合元素的元素。 $ h（\ bm {div}，ω; \ mathbb {s}）$ h（\ bm {div}）上的$ \ bm {div} \bmσ$的连续性。继承使基础功能易于计算。 $ U $的离散空间由分段$ p_ {k-2} $多项式组成，而无需任何连续性。这种混合有限元在三角形和四面体网格上以$ k \ geq 3 $稳定，并实现了最佳的收敛顺序。此外，还显示了超级融合和后处理结果。提供了一些数值实验来证明理论分析。
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 $\bmσ=-\nabla^{2}u$ is sought in the Sobolev space $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ simultaneously with the displacement $u$ in $L^{2}(Ω)$. Stemming from the structure of $H(\bm{div},Ω;\mathbb{S})$ conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ conforming finite element spaces are constructed by imposing the normal continuity of $\bm{div}\bmσ$ on the $H(\bm{div},Ω;\mathbb{S})$ 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.
