Low-regularity finite element elasticity complexes with hybridizable stresses on tetrahedral Alfeld splits

Abstract: Finite element elasticity complexes of low regularity are constructed on tetrahedral Alfeld splits. In comparison with existing three-dimensional elasticity complexes on such splits, the complexes constructed here lower both the Sobolev regularity and the polynomial degrees, while ending in a hybridizable $H({\rm div};\mathbb S)$-conforming symmetric stress space with no vertex degrees of freedom. The construction is obtained from local Bernstein-Gelfand-Gelfand arguments applied to polynomial de Rham complexes on the Alfeld split. Two local polynomial elasticity complexes are proved: an $H^{2$-$H^1({\rm} inc})$ complex and a lower-regularity $H^1({\rm curl})$-$H({\rm inc}<sup>+)$ complex. Their bubble subcomplexes and dimension formulas are derived. These local exact sequences lead to unisolvent finite elements for the displacement and incompatibility spaces and to global finite element subcomplexes of the corresponding elasticity sequences. In the lowest-order $H^1({\rm curl})$-$H({\rm inc}<sup>+)$ finite element complex, the $H({\rm inc}^+;\mathbb S)$-conforming tensor space is piecewise cubic. At the same order, the terminal stress-displacement pair recovers the Johnson-Mercier-Křížek element, while the construction covers higher-order hybridizable symmetric stresses for all $k\ge1$. A second family gives a low-regularity $H^1$-$H({\rm inc})$ finite element complex for the standard elasticity sequence for all $k\ge2$. Commuting interpolation diagrams are established for both global complexes.

• The paper presents two local exact sequences that lower Sobolev regularity and polynomial degree in elasticity complexes.
• It employs BGG mechanisms with Alfeld splits to eliminate vertex degrees of freedom and enable facet-based static condensation.
• Numerical dimension formulas and commuting interpolation diagrams ensure unisolvent finite elements and improved computational efficiency.

Low-Regularity Finite Element Elasticity Complexes with Hybridizable Stresses on Tetrahedral Alfeld Splits

Overview and Motivation

This work addresses the construction of finite element elasticity complexes of minimal regularity on three-dimensional meshes formed by Alfeld splits of tetrahedra. The goal is to lower both the Sobolev regularity and polynomial degree in the sequence of spaces leading up to a terminal symmetric stress space, while ensuring that the stress space is $H(\text{div}; S)$-conforming, admits hybridization, and has no vertex degrees of freedom.

Classically, constructing finite element elasticity complexes in three dimensions is technically challenging due to conflicting requirements: stress spaces must be both symmetric and $H(\text{div})$-conforming, leading to a proliferation of vertex-based DoFs and high polynomial orders. Furthermore, achieving conformity with respect to the incompatibility operator ($\text{inc}$) yields additional trace and regularity constraints. This work leverages the BGG (Bernstein-Gelfand-Gelfand) mechanism and polynomial de Rham complexes defined on Alfeld splits to systematically reduce regularity, eliminate vertex DoFs in the terminal stress space, and facilitate hybridization.

Background: Elasticity Complexes and Alfeld Splits

The continuous elasticity complex underlying linear elasticity in 3D is

$$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$

where $\mathrm{RM}$ denotes rigid motions, $\text{def}$ is the symmetric gradient, and $\text{inc}$ is the incompatibility operator. The precise definition of conformity in each space depends not only on bulk regularity but also on nonstandard traces, especially for $H(\text{inc},\Omega; S)$, which entails tangential-tangential and higher-order face traces.

The Alfeld split refines a tetrahedron by connecting its barycenter to its vertices, which yields a macroelement well-suited for constructing exact polynomial complexes, as in de Rham theory and elasticity. Existing complexes on Alfeld splits typically require $H^2$ or high $k$ for theoretical guarantees and include vertex DoFs in the terminal stress space, impeding hybridization and static condensation.

Contributions

The authors make several advances:

• Two Families of Local Elasticity Complexes:

The paper constructs two BGG-based, polynomial exact local complex families on each tetrahedral Alfeld split: an $H(\text{div})$0-based ($H(\text{div})$1-$H(\text{div})$2) sequence and a minimal $H(\text{div})$3-based ($H(\text{div})$4-$H(\text{div})$5) sequence.

• Reduction in Regularity and Polynomial Degree:

By utilizing hybridizable $H(\text{div})$6-conforming stress spaces (with degrees of freedom only on faces and in element interiors), both Sobolev regularity and polynomial orders in preceding spaces are reduced relative to earlier constructions.

• Hybridizable Terminal Stress Spaces:

The construction ends in a symmetric stress space that supports hybridization: post-static condensation, global stress variables are facet-based, eliminating vertex/facet coupling at the algebraic level.

• Recovery of Classical Elements at Low Order:

At lowest order, the $H(\text{div})$7-$H(\text{div})$8 sequence recovers the three-dimensional Johnson-Mercier-Křížek equilibrium element pair, ensuring practical relevance.

• Global Unisolvent Finite Elements and Commuting Diagrams:

The local complexes provide the building blocks for global finite element subcomplexes. The authors give explicit unisolvent sets of degrees of freedom for each space (including face trace and bubble DoFs), establish exactness, and prove the existence of commuting interpolation diagrams.

Technical Core

Local Complexes and Dimension Analysis

Two main local exact sequences are established:

1. $H(\text{div})$9-based ($\text{inc}$0-$\text{inc}$1) Sequence: Appropriate for higher regularity finite elements, with global conformity to $\text{inc}$2 (displacements) and $\text{inc}$3 (incompatibility).
2. $\text{inc}$4-based ($\text{inc}$5-$\text{inc}$6) Sequence: This sequence further relaxes the regularity requirements, using essentially $\text{inc}$7-conforming displacements and $\text{inc}$8-conforming symmetric tensors.

For each, precise dimension formulas (as polynomial functions of the order $\text{inc}$9) are derived for all spaces—especially significant for ensuring unisolvence and for implementation of minimal complexes.

Face and Interior Degrees of Freedom

Key to the construction are carefully designed face elements, whose DoFs are dictated by the Green's identities associated with the incompatibility operator. These face trace spaces exploit underlying two-dimensional complexes (e.g., $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$0, $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$1), which further manage the complex trace structures required for $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$2- and $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$3-conformity.

Notable Numerical and Theoretical Findings

• Dimension Reduction:

The dimension formulas show a significant reduction of total DoFs compared to prior art. For the lowest order, the terminal symmetric stress space is entirely hybridizable, with no vertex-associated DoFs, and the sequence supports $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$4 for the $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$5-based and $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$6 for the $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$7-based family.

• Bubble Subcomplexes:

The authors provide explicit constructions and dimension counts of bubble subcomplexes, which underpin the proofs of unisolvence and exactness in the full and reduced complexes.

• Commuting Interpolation:

The achieved commuting diagrams are critical for stability, robustness, and convergence analysis, ensuring the discrete complexes retain the theoretical features of the continuous models.

Implications and Perspectives

Practical Impact

The reduction of regularity and polynomial degree prerequisites directly translates into more efficient solvers and preconditioners for elasticity problems, particularly when hybridization/facet-based static condensation are employed. By eliminating global coupling at vertices, algebraic systems arising from discretization are smaller and better conditioned for modern linear solvers, which is essential for scalable algorithms in three-dimensional elasticity and related fields (e.g., defect modeling, complex material simulations).

Theoretical Outlook

The hybridizable, low-regularity complexes provide a new paradigm for structure-preserving discretization on macroelement splits. The rigorous dimension formulas and bubble arguments show how the merging of BGG theory with polynomial de Rham and elasticity complexes produces minimal, exact sequences. The ability to reproduce classical elements (Johnson-Mercier-Křížek) within a modern, robust framework further bridges the gap between legacy and advanced FE techniques.

Future Directions

Given the reduction in regularity and hybridizability, this framework is primed for extension to:

• Nonconforming and adaptive mesh settings, especially with local refinement.
• Applications in geometric elasticity or defect modeling, since the trace/compatibility conditions accommodate intrinsic and incompatibility-based models.
• High-performance implementations employing hybridized solvers and parallel static condensation.

One key avenue is extension to broader classes of topologies and to non-simplicial macroelement splits, as well as adaptation to higher-order and nonpolynomial basis functions—an area crucial for general finite element exterior calculus and higher-dimensional problems.

Conclusion

This paper advances the theory and practice of finite element elasticity complexes by producing low-regularity, hybridizable, and unisolvent finite element spaces on three-dimensional Alfeld splits. By systematically lowering Sobolev and polynomial regularity requirements, and constructing explicit commuting finite element subcomplexes ending in hybridizable $$\mathrm{RM} \rightarrow H^1(\Omega; \mathbb{R}^3) \xrightarrow{\text{def}} H(\text{inc}, \Omega; S) \xrightarrow{\text{inc}} H(\text{div}, \Omega; S) \xrightarrow{\text{div}} L^2(\Omega; \mathbb{R}^3) \rightarrow 0,$$8 symmetric stress spaces without vertex DoFs, the work resolves long-standing practical and theoretical challenges in the discretization of linear elasticity. The framework established opens new possibilities for structure-preserving, efficient, and scalable numerical methods for elasticity and related PDE systems.