论文标题
在三个维度上进行对称张量的有限元素
Finite elements for divdiv-conforming symmetric tensors in three dimensions
论文作者
论文摘要
四面体上的两种有限元元素空间是在三个维度上构造对称张量的两种有限元空间。构建的关键工具是多项式张量空间的分解和痕量运算符的表征。首先，提出了Divdiv Hilbert复合物及其相应的多项式复合物。多项式载体和张量空间的几个分解是从多项式复合物得出的。然后，通过绿色的身份来表征Div-Div操作员的痕迹。除非正常分量外，另一个涉及张量一阶衍生物组合的痕迹在整个面上是连续的。由于多项式的平稳性，对称张量元件在顶点和平面正交与每个边缘的平面也连续。第三，按照相同的方法构建了用于符合曲面的无微量张量的有限元。最后，建立了三个维度的有限元divdiv复合物以及气泡功能复合物。
Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the divdiv Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensors spaces are derived from the polynomial complexes. Then, traces for div-div operator are characterized through a Green's identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Third, a finite element for sym curl-conforming trace-free tensors is constructed following the same approach. Finally, a finite element divdiv complex, as well as the bubble functions complex, in three dimensions are established.
