H$^2-$ Korn's Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model

TL;DR

This paper introduces a new H2 Korn's inequality and its discrete form, simplifying the development of nonconforming elements for strain gradient elastic models, with proven error estimates and numerical validation.

Contribution

It establishes a novel H2 Korn's inequality and constructs new nonconforming elements with error estimates, advancing the analysis of strain gradient elastic models.

Findings

Robust nonconforming elements analyzed with convergence rates independent of material parameters

New regularized interpolation and enriching operators developed for Specht triangle and NZT tetrahedron

Numerical results confirm theoretical error estimates

Abstract

We establish a new H2 Korn's inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle [41] and the NZT tetrahedron [45] are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct new regularized interpolation estimate and the enriching operator for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results are consistent with the theoretical prediction.