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A broken Hardy inequality on finite element space and application to strain gradient elasticity

Yulei Liao, Pingbing Ming

TL;DR

This work addresses stability and convergence of finite element discretizations for strain gradient elasticity with natural boundary conditions in nearly incompressible media. It introduces a broken Hardy inequality on discontinuous FE spaces, enabling a quasi-bounded inversion of the divergence and a discrete inf-sup stable mixed FE pair whose performance is uniform in the incompressible limit and robust to the microscopic parameter $\iota$. The authors develop a nonconforming element with a tailored pressure space, prove logarithmic-factor stability, and derive error estimates that remain nearly optimal as $\iota\to 0$. Numerical experiments on polygons validate the theory, showing uniform convergence in $\iota$ and $\lambda$, optimal rates in smooth cases, and robustness in singular configurations, with implications for practical simulations on non-smooth domains.

Abstract

We illustrate a broken Hardy inequality on discontinuous finite element spaces, blowing up with a logarithmic factor with respect to the meshes size. This is motivated by numerical analysis for the strain gradient elasticity with natural boundary conditions. A mixed finite element pair is employed to solve this model with nearly incompressible materials. This pair is quasi-stable with a logarithmic factor, which is not significant in the approximation error, and converges robustly in the incompressible limit and uniformly in the microscopic material parameter. Numerical results back up that the theoretical predictions are nearly optimal. Moreover, the regularity estimates for the model over a smooth domain have been proved with the aid of the Agmon-Douglis-Nirenberg theory.

A broken Hardy inequality on finite element space and application to strain gradient elasticity

TL;DR

This work addresses stability and convergence of finite element discretizations for strain gradient elasticity with natural boundary conditions in nearly incompressible media. It introduces a broken Hardy inequality on discontinuous FE spaces, enabling a quasi-bounded inversion of the divergence and a discrete inf-sup stable mixed FE pair whose performance is uniform in the incompressible limit and robust to the microscopic parameter . The authors develop a nonconforming element with a tailored pressure space, prove logarithmic-factor stability, and derive error estimates that remain nearly optimal as . Numerical experiments on polygons validate the theory, showing uniform convergence in and , optimal rates in smooth cases, and robustness in singular configurations, with implications for practical simulations on non-smooth domains.

Abstract

We illustrate a broken Hardy inequality on discontinuous finite element spaces, blowing up with a logarithmic factor with respect to the meshes size. This is motivated by numerical analysis for the strain gradient elasticity with natural boundary conditions. A mixed finite element pair is employed to solve this model with nearly incompressible materials. This pair is quasi-stable with a logarithmic factor, which is not significant in the approximation error, and converges robustly in the incompressible limit and uniformly in the microscopic material parameter. Numerical results back up that the theoretical predictions are nearly optimal. Moreover, the regularity estimates for the model over a smooth domain have been proved with the aid of the Agmon-Douglis-Nirenberg theory.

Paper Structure

This paper contains 16 sections, 21 theorems, 174 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A broken Hardy inequality on finite element spaces
  3. Application for strain gradient elasticity
  4. The mixed variational formulation
  5. A nonconforming finite element
  6. The mixed finite element approximation
  7. Error estimates
  8. Numerical examples
  9. The first example
  10. The second example
  11. The third example
  12. The fourth example
  13. Conclusion
  14. Regularity estimates on the smooth domain
  15. Lower regularity estimates
  16. ...and 1 more sections

Key Result

Theorem 2.1

There exists a constant $C$ depending on $\sigma_1,\sigma_2,d,r$ but independent of $h$ such that for sufficient small $h>0$, there holds for any $f\in L^d(B_1^d)$ satisfying $f|_K\in\mathbb{P}_r(K)$ for all $K\in\mathcal{T}_h(B_1^d\backslash\{\boldsymbol{0}\})$ and $f(\boldsymbol{0})=0$, with a nonnegative integer $r$ and $\rho=\lvert\boldsymbol{x}\rvert$.

Figures (2)

  • Figure 1: Diagram for the proof of the broken Hardy inequality when $d=2$
  • Figure 2: Diagram for the degrees of freedom.

Theorems & Definitions (41)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • Lemma 3.2: Arnold:1988
  • Lemma 3.3
  • Lemma 3.4
  • ...and 31 more