论文标题
复合物的复合物：三个维度的有限元复合物
Complexes from Complexes: Finite Element Complexes in Three Dimensions
论文作者
论文摘要
在求解部分微分方程（PDE）的领域，希尔伯特复合物已变得非常重要。最近的进步着重于使用Arnold和Hu [复合物中的复合物所示的Bernstein-Gelfand-Gelfand（BGG）框架创建新的复合物。 {\ em发现。计算。数学。}，2021]。本文将其方法扩展到三维有限元复合物。有限元Hessian，弹性和Divdiv复合物是通过应用诸如平滑有限元de rham复合物，$ t $ - $ n $分解以及跟踪复合物以及相关二维有限元元素类似物等技术来系统地得出的。该构建包括两个还原操作和一个扩展操作，以解决BGG图中的连续性差异，最终导致了一个构建有限元复合物的全面有效框架，这些元素复合物在PDE求解中具有各种应用。
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.
