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

Jul 16, 2026

Finite element elasticity complexes of low regularity are constructed on tetrahedral Alfeld splits. Compared with existing three-dimensional elasticity complexes on these splits, the new complexes reduce Sobolev regularity and polynomial degrees, while ending in a hybridizable H(div;S)-conforming symmetric stress space with no vertex degrees of freedom. The construction uses local Bernstein–Gelfand–Gelfand arguments on polynomial de Rham complexes and yields two local exact elasticity complexes, bubble subcomplexes, and dimension formulas leading to unisolvent elements and commuting interpolation diagrams.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages23

Reading Time58 min

Introduction

Elasticity complexes and function spaces

Challenges in constructing subcomplexes

Related work and predecessors

arXiv :2607 .01933v1 [math .NA] 2 Jul 2026 LOW-REGULARITY FINITE ELEMENT ELASTICITY COMPLEXES WITH HYBRIDIZABLE STRESSES ON TETRAHEDRAL ALFELD SPLITS JOHNNY GUZM´AN AND XUEHAI HUANG 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(div; 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 (inc) complex and a lower-regularity H 1 (curl)–H(inc+ ) 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 (curl)–H(inc+ ) finite element complex, the H(inc+ ; S)-conforming tensor space is piecewise cubic. At the same order, the terminal stress–displacement pair recovers the Johnson–Mercier–Kˇr´ıˇzek element, while the construction covers higher-order hybridizable symmetric stresses for all k ≥ 1. A second family gives a low-regularity H 1–H(inc) finite element complex for the standard elasticity sequence for all k ≥ 2. Commuting interpolation diagrams are established for both global complexes. 1. Introduction Finite element complexes provide a structural framework for constructing conforming finite element spaces whose unknowns are linked by differential operators. For linear elasticity in three space dimensions, the relevant continuous complex is (1 . 1) RM H 1 (Ω;R3 ) f H(inc, Ω;S) c H(div, Ω;S) i L2 (Ω;R3 ) → 0 , where RM is the space of infinitesimal rigid motions, def = sym grad is the linearized strain, and inc is the incompatibility operator. The tensor-valued Sobolev spaces in (1.1) are H(inc, Ω;S) := {τ ∈ L2 (Ω;S) : inc τ ∈ L2 (Ω;S)} , H(div, Ω;S) := {τ ∈ L2 (Ω;S) : div τ ∈ L2 (Ω;R3 )} . Thus H(inc, Ω;S) is the space for symmetric tensor fields with square-integrable incompatibility, while H(div, Ω;S) is the natural space for symmetric stress tensors. The complex is the linear elasticity analogue ofthe de Rham complex and plays an important role in mixed elasticity and structure-preserving discretizations [4, 5], intrinsic elasticity and Saint-Venant compatibility conditions [20, 18], and models of defects and incompatibility [29, 1] . It also gives explicit descriptions of kernels and ranges, which are useful in stability analysis, preconditioning, and the construction of commuting projections; see, for example,[5, 16, 15] . Constructing finite element subcomplexes of (1.1) is delicate for two related reasons. First, the stress space must enforce both symmetry and H(div) conformity. Classical polynomial symmetric stress elements are stable, but they typically involve vertex degrees of freedom and relatively high polynomial degrees. Second, the preceding H(inc) space has nonstandard traces: tangential–tangential components and second-order surface differential information enter the Green identity for the incompatibility operator. Consequently, conforming H(inc) elements are substantially more constrained than standard H(curl)-or H(div)-conforming elements. The literature contains both constructions of stable symmetric stress spaces and constructions of full finite element elasticity complexes. On simplicial meshes, the two-dimensional Arnold–Winther element [7], together with its interpretation through finite element exterior calculus and the Bernstein– Gelfand–Gelfand (BGG) construction [5], gives a conforming discretization of the elasticity complex on triangular meshes. More systematic tw

FAQs

What is constructed in the paper, and on what meshes?

The paper constructs low-regularity finite element elasticity complexes on tetrahedral Alfeld splits.

How does the proposed approach differ from existing 3D elasticity complexes on Alfeld splits?

It lowers Sobolev regularity and polynomial degrees while using a hybridizable H(div;S)-conforming symmetric stress space without vertex degrees of freedom.

Which key tools and results are used to build the complexes?

Local Bernstein–Gelfand–Gelfand arguments applied to polynomial de Rham complexes produce local exact sequences, bubble subcomplexes, dimension formulas, and commuting interpolation diagrams, leading to unisolvent finite elements.

Jul 16, 2026

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