  This paper presents the first family of conforming finite element divdiv
complexes on tetrahedral grids in three dimensions. In these complexes, finite
element spaces of $H(\text{divdiv},\Omega;\mathbb{S})$ are from a current
preprint [Chen and Huang, arXiv: 2007.12399, 2020] while finite element spaces
of both $H(\text{symcurl},\Omega;\mathbb{T})$ and $H^1(\Omega;\mathbb{R}^3)$
are newly constructed here. It is proved that these finite element complexes
are exact. As a result, they can be used to discretize the linearized
Einstein-Bianchi system within the dual formulation.
