Complexes from Complexes: Finite Element Complexes in Three Dimensions

TL;DR

This paper extends the BGG framework to systematically construct three-dimensional finite element complexes, including Hessian, elasticity, and divdiv complexes, enhancing PDE solving techniques.

Contribution

It introduces a comprehensive method for deriving 3D finite element complexes using advanced techniques like smooth de Rham complexes and trace complexes, building on prior BGG approaches.

Findings

Systematic derivation of 3D finite element Hessian, elasticity, and divdiv complexes.

Application of reduction and augmentation operations to handle continuity issues.

Framework applicable to various PDE solving scenarios.

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.