The linearized Einstein–Bianchi (LEB) system is derived from the vacuum Einstein equations where the Ricci tensor vanishes and consequently leads to the reduction of the Riemann tensor to the Weyl tensor. Following the Bel decomposition with respect to timelike unit vectors, the Weyl tensor can be separated into electric and magnetic components. Due to the second Bianchi identity, these linearized electric and magnetic tensors obey Maxwell-type equations, analogous to those governing the electric and magnetic field vectors. But an essential difference is that
the unknowns E and B are a symmetric and traceless matrix, rather than vectors. This distinction presents a significant challenge in constructing stable finite elements. This talk constructs the finite element Gradgrad and Divdiv complexes on tetrahedral grids, which is exact if the domain is contractible and Lipschitz. These finite element complexes provide structure- preserving numerical methods for the LEB system.