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

This paper presents the first family of conforming finite element complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of H(,Ω;) are from a recent article [L. Chen and X. Huang, Math. Comp., 91 (2022), pp. 1107--1142] while finite element spaces of both H(,Ω;) and H¹(Ω;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.