Complexes from Complexes: Finite Element Complexes in Three Dimensions

In the field of solving partial differential equations (PDEs), Hilbert complexes have become highly significant. Recent advances focus on creating new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as shown by Arnold and Hu [Complexes from complexes. {\em Found. Comput. Math.}, 2021]. This paper extends their approach to three-dimensional finite element complexes. The finite element Hessian, elasticity, and divdiv complexes are systematically derived by applying techniques such as smooth finite element de Rham complexes, the $t$-$n$ decomposition, and trace complexes, along with related two-dimensional finite element analogs. The construction includes two reduction operations and one augmentation operation to address continuity differences in the BGG diagram, ultimately resulting in a comprehensive and effective framework for constructing finite element complexes, which have various applications in PDE solving.