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

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 $C<sup>1$ 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 $C<sup>1$ finite elements on barycentric refinements, including quadratic, cubic, quartic, and higher-order elements.

• The paper introduces explicit closed-form formulas for enrichment functions, establishing a planar finite element elasticity complex with optimal approximation properties.
• It constructs C^1 potential spaces via Airy potentials, enabling direct computation and minimal local degrees for stable mixed finite element formulations.
• The approach simplifies global assembly by avoiding C^2 vertex conditions and compares favorably to conventional methods like the ADG and Arnold–Winther elements.

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

Introduction and Context

This work addresses the constructive realization of the two-dimensional elasticity exact sequence for mixed finite element spaces tailored to plane elasticity on barycentric refinements. The paper makes explicit the sequence underpinning the Arnold--Douglas--Gupta (ADG) family of higher-order mixed elements, providing closed-form formulas for the associated enrichment functions previously only available algorithmically. By formulating explicit Airy potentials, this enables a completely concrete Hsieh--Clough--Tocher-type ($C^1$) potential space whose image under the Airy operator exactly coincides with the ADG stress space. The main outcome is an explicit global finite element elasticity complex with optimal approximation properties, minimal local dimension, and straightforward computability, supporting quadratic and higher-order finite element constructions.

Construction of Stress and Potential Spaces

The finite element discretizations for plane elasticity require symmetric $H(\text{div})$-conforming stress spaces and compatible displacement or potential spaces. The ADG construction provides $\Sigma_{k,h}^{\text{ADG}}$ by enriching $\mathbb{P}_k(T;\mathbb{S})$ on barycentric refinements with three locally supported, divergence-free stresses, ensuring the polynomial character of the divergence over macro elements. The authors derive explicit symmetric enrichment functions $\psi_i$ (for $i=0,1,2$), each supported on a macro triangle $T$ and obtained in barycentric coordinates, satisfying $\operatorname{div} \psi_i = 0$. These are shown to admit $C^1$ potential functions $C^1$0 such that $C^1$1, with $C^1$2 the Airy (rotated Hessian) operator.

Figure 1: The lowest-order elasticity complex, $C^1$3--$C^1$4--$C^1$5, on a macrotriangle and its barycentric subdivision.

The element enrichment is verified to be direct, and unisolvence of the edge and interior moment degrees of freedom is proven. The global space $C^1$6 is assembled by enforcing continuity of normal fluxes on macro edges. These explicit basis constructions avoid the need for higher continuity in the scalar potential (such as the $C^1$7 at vertices of the Argyris element), thereby simplifying both implementation and assembly of global complexes.

Degrees of Freedom and Basis Construction

The associated $C^1$8 scalar potential spaces $C^1$9 are structured as $H(\text{div})$0 plus the span of the three explicit potentials $H(\text{div})$1. Degrees of freedom include function and gradient values at triangle vertices, edge moments (function and normal derivative), and select interior moments, designed to achieve precise unisolvence. A block lower triangular structure results in dof/basis systems, ensuring efficient computation and assembly.

Figure 2: Degrees of freedom for the $H(\text{div})$2 elements $H(\text{div})$3, $H(\text{div})$4, and $H(\text{div})$5 (dimensions 18, 12, and 9, respectively).

The authors further offer explicit, dual local bases and clarify the hierarchical relationship between different polynomial degree elements, including $H(\text{div})$6 (the cubic HCT) and its subspace $H(\text{div})$7. The normal derivative conformity is obtained with the minimal additional enrichment (avoiding the extra vertex conditions present in the Argyris setup).

Geometrical Interpretation and Barycentric Refinement

The use of barycentric refinement introduces additional local structure, reflected in the geometric degrees of freedom and basis. The red lattice points of standard HCT elements cannot be assigned to two edges simultaneously; in this construction, the enrichment introduces blue lattice points that can, enabling the normal derivatives required for global $H(\text{div})$8 conformity without auxiliary conditions.

Figure 3: Barycentric refinement of a triangle used to realize explicit basis enrichment.

Figure 4: Duplicated lattice points (red) in refinement; their partitioning (blue) enables enough degrees of freedom for the normal derivative.

Exact Sequence and Commuting Diagram

The scalar potential, stress, and displacement spaces fit together into an exact finite element elasticity sequence:

$H(\text{div})$9

with commuting interpolation operators. The lowest-order instance, $\Sigma_{k,h}^{\text{ADG}}$0--$\Sigma_{k,h}^{\text{ADG}}$1--$\Sigma_{k,h}^{\text{ADG}}$2, is fully explicit and minimal in local dimension, while the same framework scales to higher degrees.

The construction ensures that $\Sigma_{k,h}^{\text{ADG}}$3 and the associated inf-sup stability condition holds uniformly. The abstract Hilbert complex structure is mirrored discretely, and global quasi-interpolation operators are defined to respect the sequence and preserve commutativity, a crucial property for stability and convergence analysis in mixed formulations.

Comparison and Implications

This explicit realization combines the advantages of several approaches. Compared to the Arnold--Winther and Hu--Zhang elements, the new construction avoids $\Sigma_{k,h}^{\text{ADG}}$4 vertex requirements, instead only enforcing $\Sigma_{k,h}^{\text{ADG}}$5, and dispenses with global stabilization mechanisms of virtual element methods on triangles. In contrast to standard composite $\Sigma_{k,h}^{\text{ADG}}$6 elements, the enrichment is minimal: only three explicit shape functions are needed per macro element, yielding lower local dimension and considerably simplifying implementation.

The result is a family of $\Sigma_{k,h}^{\text{ADG}}$7 elements supporting direct calculation without recourse to computer algebra or stabilization, with immediate applications in conforming $\Sigma_{k,h}^{\text{ADG}}$8 and mixed $\Sigma_{k,h}^{\text{ADG}}$9-based discretizations for elasticity, biharmonic, and related PDEs.

Potential Extensions and Future Directions

The explicit recipe for constructing $\mathbb{P}_k(T;\mathbb{S})$0 finite elements and an exact finite element elasticity complex can be extended to other mesh structures that admit similar enrichment mechanics. The authors' barycentric-mesh-specific strategy invites parallel developments for polygonal $\mathbb{P}_k(T;\mathbb{S})$1 virtual elements, possibly leading to a serendipity family with reduced interior degrees of freedom. The general methodology—explicit enrichment for exactness, minimal unisolvence—could yield advances in hybrid-mixed, multi-physics, and high-order compatible finite elements, particularly where explicit expressions facilitate analysis and implementation.

Conclusion

The paper provides a full explicit construction of the ADG-type finite element elasticity complex on planar barycentric refinements, including closed formulas for local enrichment, explicit $\mathbb{P}_k(T;\mathbb{S})$2 potential spaces, and practical global assembly strategies. The approach yields new, minimal-dimension, stable families of $\mathbb{P}_k(T;\mathbb{S})$3 elements for elasticity, significantly clarifying the algebraic and geometric structure of these complexes and inviting further generalization to more involved mesh and domain types.