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

Complexes from Complexes: Finite Element Complexes in Three Dimensions