Finite Element Complexes in Two Dimensions

TL;DR

This paper systematically constructs various two-dimensional finite element complexes, including de Rham, elasticity, and divdiv complexes, introducing new tools like lattice decomposition and BGG framework for advanced finite element analysis.

Contribution

It provides the first systematic construction of finite element complexes in 2D, introducing novel tools like lattice decomposition and discrete BGG for finite element theory.

Findings

Constructed finite element complexes with various smoothness levels.

Developed smooth scalar finite elements based on lattice decomposition.

Derived elasticity and divdiv complexes using the BGG framework.

Abstract

In this study, two-dimensional finite element complexes with various levels of smoothness, including the de Rham complex, the curldiv complex, the elasticity complex, and the divdiv complex, are systematically constructed. Smooth scalar finite elements in two dimensions are developed based on a non-overlapping decomposition of the simplicial lattice and the Bernstein basis of the polynomial space, with the order of differentiability at vertices being greater than twice that at edges. Finite element de Rham complexes with different levels of smoothness are devised using smooth finite elements with smoothness parameters that satisfy certain relations. Finally, finite element elasticity complexes and finite element divdiv complexes are derived from finite element de Rham complexes by using the Bernstein-Gelfand-Gelfand (BGG) framework. This study is the first work to construct finite element complexes in a systematic way. Moreover, the novel tools developed in this work, such as the non-overlapping decomposition of the simplicial lattice and the discrete BGG construction, can be useful for further research in this field.