Mixed Finite element method for stress gradient elasticity

TL;DR

This paper introduces stable finite element methods for stress gradient elasticity, effectively capturing size effects and providing robust error estimates, with numerical validation confirming optimal convergence.

Contribution

It develops novel finite element pairs with proven stability and error bounds for stress gradient elasticity, addressing limitations of classical models.

Findings

Stable finite element pairs achieve unconditional stability with higher vertex continuity.

Error estimates are robust and independent of model parameters.

Numerical experiments confirm optimal convergence rates.

Abstract

This paper develops stable finite element pairs for the linear stress gradient elasticity model, overcoming classical elasticity's limitations in capturing size effects. We analyze mesh conditions to establish parameter-robust error estimates for the proposed pairs, achieving unconditional stability for finite elements with higher vertex continuity and conditional stability for Continuous Galerkin-Discontinuous Galerkin (CG-DG) pairs when no interior vertex has edges lying on three or fewer lines. Numerical experiments validate the theoretical results, demonstrating optimal convergence rates.