论文标题
低度稳健的Hellinger-Reissner-Reissner有限元方案，用于平面线性弹性，并具有对称应力张量
Low-degree robust Hellinger-Reissner finite element schemes for planar linear elasticity with symmetric stress tensors
论文作者
论文摘要
在本文中，我们研究了低度稳健有限元方案的构建，以实现一般三角形的平面线性弹性。 首先，我们提出了一个低度不合格的Helling-Reissner有限元方案。对于应力张量空间，分段多项式形状函数空间为$$ {\ rm span} \ left \ {\ left（\ begin {array} {ccc} {cc} 1＆0 \ \ 0＆0＆0＆0 \ end {arnay} 0 \ end {array} \ right），\ left（\ begin {array} {cc} 0＆0＆0 \\ 0＆1 \ end {array} \ right），\ left（\ begin {array} 0 \ end {array} \ right），\ left（\ begin {arnay} {cc} 0＆y \\ y＆0 \ y＆0 \ end {array} \ right），\ left（\ begin {arnay} {array} {cc} {cc} {cc} 0＆x^2-y^2-y^2-y^2 y^2 y^2 y^2 y^2 y^y^2 y \ $ nary}总空间的尺寸渐近地是顶点数量的8次，基本函数的支撑是每个边缘的斑块。分段刚体空间用于位移。列出了$ \ mathbb {l}^2 $中的强大错误估计和$ \ mathbf {h}（{\ rm div}）$ norms。 其次，我们研究了具有最低程度多项式函数空间的方案的理论结构。具体来说，构建了Hellinger-Reissner的有限元方案，其局部形状功能空间是压力张量为5维的，这对于$ \ Mathbf {H}的局部近似值是最低的，（{\ rm div}; \ Mathbb {s}）$，以及用于置换构成的空间。 $ \ mathbb {l}^2 $中的强大错误估计和$ \ MATHBF {h}（{\ rm div}）$ narms用于常规解决方案和数据。
In this paper, we study the construction of low-degree robust finite element schemes for planar linear elasticity on general triangulations. Firstly, we present a low-degree nonconforming Helling-Reissner finite element scheme. For the stress tensor space, the piecewise polynomial shape function space is $$ {\rm span}\left\{\left(\begin{array}{cc}1&0\\ 0 & 0\end{array}\right),\left(\begin{array}{cc}0&1\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}0&0\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}0&x\\ x & 0\end{array}\right),\left(\begin{array}{cc}0&y\\ y & 0\end{array}\right),\left(\begin{array}{cc}0&x^2-y^2\\ x^2-y^2 & 0\end{array}\right)\right\}, $$ the dimension of the total space is asymptotically 8 times of the number of vertices, and the supports of the basis functions are each a patch of an edge. The piecewise rigid body space is used for the displacement. Robust error estimations in $\mathbb{L}^2$ and broken $\mathbf{H}({\rm div})$ norms are presented. Secondly, we investigate the theoretical construction of schemes with lowest-degree polynomial shape function spaces. Specifically, a Hellinger-Reissner finite element scheme is constructed, with the local shape function space for the stress tensor being 5-dimensional which is of the lowest degree for the local approximation of $\mathbf{H}({\rm div};\mathbb{S})$, and the space for the displacement is piecewise constants. Robust error estimations in $\mathbb{L}^2$ and broken $\mathbf{H}({\rm div})$ norms are presented for regular solutions and data.
