Explicit Planar Finite Element Elasticity Complexes and $C^1$ Elements on Barycentric Refinements

TL;DR

This paper explicitly constructs finite element elasticity complexes on barycentric refinements, introducing new $C^1$ elements of various orders with clear formulas and potential spaces.

Contribution

It makes the exact-sequence structure of Arnold--Douglas--Gupta elements explicit and develops a new family of $C^1$ finite elements on barycentric refined meshes.

Findings

Derived closed-form formulas for stress space enrichments.

Identified explicit Airy potentials generating the stress spaces.

Constructed new $C^1$ finite elements of quadratic to higher orders.

Abstract

The exact-sequence structure behind the Arnold--Douglas--Gupta family of higher-order mixed finite elements for plane elasticity on barycentric refinements is made explicit. On each macro triangle, the symmetric stress space is obtained by enriching polynomial stresses with three locally supported functions. We derive closed-form formulas for these enrichments and identify explicit Airy potentials that generate them. This leads to a concrete Hsieh--Clough--Tocher type $C^1$ potential space whose Airy image is exactly the Arnold--Douglas--Gupta stress space. By enforcing single-valued degrees of freedom, we obtain global spaces and a fully explicit finite element elasticity complex on simply connected domains. As a consequence, we construct a new family of $C^1$ finite elements on barycentric refinements, including quadratic, cubic, quartic, and higher-order elements.