  In the realm of solving partial differential equations (PDEs), Hilbert
complexes have gained paramount importance, and recent progress revolves around
devising new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as
demonstrated by Arnold and Hu [Complexes from complexes. {\em Found. Comput.
Math.}, 2021]. This paper significantly extends this methodology to
three-dimensional finite element complexes, surmounting challenges posed by
disparate degrees of smoothness and continuity mismatches. By incorporating
techniques such as smooth finite element de Rham complexes, the $t-n$
decomposition, and trace complexes with corresponding two-dimensional finite
element analogs, we systematically derive finite element Hessian, elasticity,
and divdiv complexes. Notably, the construction entails the incorporation of
reduction operators to handle continuity disparities in the BGG diagram at the
continuous level, ultimately culminating in a comprehensive and robust
framework for constructing finite element complexes with diverse applications
in PDE solving.
