Complexes from Complexes: Finite Element Complexes in Three Dimensions

TL;DR

This work develops a unified three-dimensional finite element framework for Hilbert complexes using the Bernstein–Gelfand–Gelfand construction. By combining smooth finite element de Rham complexes, a $t$-$n$ decomposition, and trace/bubble complexes, the authors derive FE Hessian, elasticity, and divdiv complexes, addressing continuity mismatches via two reduction operations and an augmentation. The results include explicit constructions of face and edge conforming elements, div/curl stability analyses, and multiple variants of divdiv complexes, enabling robust discretizations for PDEs in 3D. The framework not only unifies prior 2D results but also provides a systematic methodology for building new FE complexes with controlled smoothness and stability, with potential impact on elasticity, plate models, biharmonic problems, and related PDE solvers.

Abstract

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.

Figures (6)

• Figure 1: Organization of Sections 3 - 6.
• Figure 2: Two reduction operations. The $\widetilde{\quad}$ operation reduces the space $V_1$ to $\widetilde{V}_1$ and $V_2$ to $\widetilde{V}_2$ by removing the sub-spaces marked by the light gray color. The $\widehat{\quad}$ operation reduces the space $V_0$ to $\widehat{V}_0$ and $V_1$ to $\widehat{V}_1$ by removing the sub-spaces marked by the dark gray color. The inverse hat operation is adding these sub-spaces back.
• Figure 3: The $t$-$n$ decompositions of $\mathbb T$ on edges and faces. Red blocks are associated to bubbles and green blocks for the normal traces which are redistributed to faces.
• Figure 4: The $t$-$n$ decomposition of $\mathbb S$ on edges and faces. Red blocks are associated to bubbles, and green and blue blocks for the normal traces. The green blocks can be redistributed to faces while the blue blocks introduces stronger continuity on the normal plane $\mathscr N^e(\mathbb S)$.
• Figure 5: Diverse configurations of finite element divdiv complexes. The fundamental finite element divdiv complex \ref{['eq:femdivdivcomplex3d+']}, depicted by white blocks, is formulated through the BGG framework, with additional smoothness. Then add green blocks to get \ref{['eq:femdivdivcomplex3dvariant1']}, add red blocks to get \ref{['eq:femdivdivcomplex3dvariant2']}, and add both to get \ref{['eq:femdivdivcomplex3d']}.

Theorems & Definitions (110)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• Lemma 3.2: Theorem 4.5 in Chen;Huang:2022FEMcomplex3D
• Lemma 3.3
• Lemma 3.4: Theorem 4.10 in Chen;Huang:2022FEMcomplex3D