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.

翻译：在求解偏微分方程（PDE）领域，希尔伯特复形已具有至关重要的地位，而最新进展主要围绕利用Bernstein-Gelfand-Gelfand（BGG）框架设计新型复形展开，正如Arnold与Hu在《复形构造新法》（Complexes from complexes，《计算数学基础》2021年刊）中所论证。本文将该方法显著拓展至三维有限元复形，克服了由光滑性差异与连续性不匹配带来的挑战。通过引入光滑有限元de Rham复形、$t-n$分解以及迹复形（含对应二维有限元类比）等技术手段，我们系统推导出有限元Hessian复形、弹性复形与divdiv复形。值得注意的是，该构造过程需融入约化算子以处理连续层面BGG图中的连续性差异，最终构建出通用且稳健的有限元复形框架，可广泛用于PDE求解。