We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain gradcurl-conforming finite elements and inf-sup stable Stokes pairs on tetrahedra which fit into complexes. We show that the new elements lead to convergent algorithms for solving a gradcurl model problem as well as solving the Stokes system with precise divergence-free condition. We demonstrate the validity of the algorithms by numerical experiments.
Design and homological analysis of twisted and BGG Stokes-de Rham complexes on polygonal meshes
IMAG, Univ. Montpellier, CNRS（蒙彼利埃大学 IMAG 研究中心）；School of Mathematics, Monash University（莫纳什大学数学学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff--Love plates
IMAG, Univ. Montpellier, CNRS（蒙彼利埃大学，法国国家科学研究中心）；School of Mathematics, Monash University（莫纳什大学数学学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）
Conforming finite element DIVDIV complexes and the application for the linearized Einstein-Bianchi system
本文给出了三维四面体网格上的第一类协调有限元divdiv复形。在这些复形中，$H(\text{divdiv},Ω;\mathbb{S})$的有限元空间来自当前的预印本[Chen and Huang，arXiv：2007.12399，2020]，而$H(\text{symcurl},Ω;\mathbb{T})$和$H^1(Ω;\mathbb{R}^3)$的有限元空间都是新构造的。证明了这些有限元复形是精确的。因此，他们可以用来离散的线性Einstein-Bianchi系统的对偶制定。
