Explicit Planar Finite Element Elasticity Complexes and $C^1$ Elements on Barycentric Refinements
For researchers in computational mechanics and finite element methods, this provides a fully explicit finite element elasticity complex and new C^1 elements, but the work is incremental as it builds on existing frameworks.
The paper makes explicit the exact-sequence structure of the Arnold-Douglas-Gupta mixed finite elements for plane elasticity on barycentric refinements, deriving closed-form enrichments and Airy potentials. This yields a new family of C^1 finite elements (quadratic, cubic, quartic, and higher-order) on barycentric refinements.
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.