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Uniformly $hp$-stable elements for the elasticity complex

Francis R. A. Aznaran, Kaibo Hu, Charles Parker

TL;DR

This work develops an hp-stable discretization framework for the elasticity complex with symmetric stress in 2D using the Hu–Zhang element. It provides a constructive, $h$- and $p$-uniform right inverse of the divergence via polynomial-preserving Poincaré operators in the FEEC/BGG setting, and builds hp-bounded commuting projections together with hp-stable Hodge decompositions. The results extend to the Arnold–Winther family and are supported by numerical experiments demonstrating robust stability across mesh refinements and high polynomial degrees. Collectively, the paper delivers a rigorous, implementable approach for hp-robust symmetric-stress discretizations in linear elasticity with mixed boundary conditions.

Abstract

For the discretization of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu--Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein--Gelfand--Gelfand framework of the finite element exterior calculus. We also construct $hp$-bounded projection operators satisfying a commuting diagram property and $hp$-stable Hodge decompositions. Numerical examples are provided.

Uniformly $hp$-stable elements for the elasticity complex

TL;DR

This work develops an hp-stable discretization framework for the elasticity complex with symmetric stress in 2D using the Hu–Zhang element. It provides a constructive, - and -uniform right inverse of the divergence via polynomial-preserving Poincaré operators in the FEEC/BGG setting, and builds hp-bounded commuting projections together with hp-stable Hodge decompositions. The results extend to the Arnold–Winther family and are supported by numerical experiments demonstrating robust stability across mesh refinements and high polynomial degrees. Collectively, the paper delivers a rigorous, implementable approach for hp-robust symmetric-stress discretizations in linear elasticity with mixed boundary conditions.

Abstract

For the discretization of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu--Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein--Gelfand--Gelfand framework of the finite element exterior calculus. We also construct -bounded projection operators satisfying a commuting diagram property and -stable Hodge decompositions. Numerical examples are provided.
Paper Structure (35 sections, 25 theorems, 145 equations, 5 figures)

This paper contains 35 sections, 25 theorems, 145 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem setting and the elasticity complex
  3. Defining the complex with boundary conditions
  4. Properties of the continuous elasticity complex
  5. Finite element spaces
  6. Local degrees of freedom for the stress space
  7. Global degrees of freedom for the stress space
  8. Summary of main results
  9. Uniformly stable inversion of the divergence operator and consequences
  10. $hp$-bounded commuting cochain projections
  11. $hp$-stable Hodge decompositions
  12. Extension to the Arnold-Winther space
  13. Numerical examples
  14. Bounded Poincaré operators on an element
  15. Inverting divergence without boundary conditions
  16. ...and 20 more sections

Key Result

Theorem 2.2

\newlabelthm:h1-inversion-div-bc0 ($H^1(\Omega; \mathbb{S})$ inversion of the divergence.) For every $u \in L^2_{\Gamma}(\Omega; \mathbb{V})$, there exists $\sigma \in H^1(\Omega; \mathbb{S}) \cap H_{\Gamma}(\mathop{\mathrm{div}}\nolimits, \Omega; \mathbb{S})$ such that where $C$ is independent of $u$.

Figures (5)

  • Figure 1: Example domain $\Omega$ and boundary partition $\mathcal{D} = \{2, 3, 6, 7, 9, 10, 11, 12\}, \mathcal{N} = \{1, 4, 5, 8\}$.
  • Figure 1: Complex containing the Hu--Zhang elements Hu2014 at lowest order.
  • Figure 1: Computational domains
  • Figure 1: Notation for (a) general triangle $K$ and (b) reference triangle $\hat{K}$.
  • Figure 2: Inf-sup constants $\beta(h, p)$ for the Hu--Zhang pair $\Sigma_\Gamma^p\times V_\Gamma^{p - 1}$ for various polynomial degrees $p$ and on various meshes.

Theorems & Definitions (42)

  • Remark 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Lemma 3.3
  • Proof 3
  • Lemma 3.4
  • ...and 32 more