Conforming Finite Element DIVDIV Complexes and the Application for the Linearized Einstein--Bianchi System

This paper presents the first family of conforming finite element $\ddiv\ddiv$ complexes on tetrahedral grids in three dimensions. In these complexes, spaces $H(\ddiv\ddiv,\Omega;\Sbb)$ are from a recent article [L. Chen and X. Huang, Math. Comp., 91 (2022), pp. 1107--1142] while both $H(\sym\ccurl,\Omega;\Tbb)$ $H^1(\Omega;\R^3)$ newly constructed here. It is proved that exact. As result, they can be used to discretize linearized Einstein--Bianchi system within dual formulation.