Hybridizable Symmetric Stress Elements on the Barycentric Refinement in Arbitrary Dimensions

TL;DR

This work develops hybridizable $H(\operatorname{div})$-conforming finite elements for symmetric tensor fields on simplices in arbitrary dimensions, leveraging barycentric refinement and a tangential-normal decomposition to redistribute degrees of freedom while preserving symmetry and stability. The authors introduce a family of enriched spaces on the refined mesh, including $\Sigma_{k,\phi}^{\operatorname{div}}$, $\Sigma_{k,\phi,nn}^{\operatorname{div}}$, and related reduced variants, and prove inf-sup stability for these spaces both on the refined and coarse meshes. High-order elements are built via a $t$-$n$ decomposition with bubble enrichments that enable face-wise DoF redistribution and allow RT-type and even more robust div-conforming behavior. A stabilized mixed formulation for linear elasticity is analyzed, and a hybridization strategy based on a weak div operator yields a solvable, scalable system with a postprocessing step that achieves superconvergence for the displacement. Overall, the approach provides flexible, dimension-agnostic, high-order, hybridizable elements for symmetric stresses with robust elasticity discretizations independent of the Lamé parameter.

Abstract

Hybridizable \(H(\textrm{div})\)-conforming finite elements for symmetric tensors on simplices with barycentric refinement are developed in this work for arbitrary dimensions and any polynomial order. By employing barycentric refinement and an intrinsic tangential-normal (\(t\)-\(n\)) decomposition, novel basis functions are constructed to redistribute degrees of freedom while preserving \(H(\textrm{div})\)-conformity and symmetry, and ensuring inf-sup stability. These hybridizable elements enhance computational flexibility and efficiency, with applications to mixed finite element methods for linear elasticity.

Figures (3)

• Figure 1: Barycentric refinement.
• Figure 2: Red block ($\circ$): $H(\operatorname{div})$-bubble polynomial basis. Green ($\diamond$): redistributed basis. Blue ($\hbox{$\square$}$): basis of $\mathbb S(\mathscr N^f)$ with a global normal plane basis. Here, $f$ is an edge of a tetrahedron, so $\dim \mathscr N^f = 2$. The vectors $\{\boldsymbol{n}_{1}^{f}, \boldsymbol{n}_{2}^{f}\}$ in (a) form a global basis of $\mathscr N^f$ used to impose the symmetric constraints. With the enrichment, we may instead use the face normals $\{\boldsymbol{n}_{F_{1}}, \boldsymbol{n}_{F_{2}}\}$, as shown in (b), so that the DoFs are redistributed to the two faces $F_1$ and $F_2$ containing $f$.
• Figure :

Theorems & Definitions (66)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof