Robust a posteriori error estimators for mixed approximation of nearly incompressible elasticity

TL;DR

This paper develops and analyzes robust a posteriori error estimators for mixed finite element methods applied to nearly incompressible linear elasticity, ensuring accuracy independent of material incompressibility.

Contribution

It introduces novel error estimators with proven robustness in the incompressible limit for mixed elasticity problems.

Findings

Error estimators are reliable and efficient across nearly incompressible regimes.

Theoretical bounds are independent of Lamé coefficients, confirming robustness.

Numerical experiments validate the theoretical error estimates.

Abstract

This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori error estimators for the energy norm of the finite element error are proposed and analysed. We establish upper and lower bounds for the energy error in terms of the proposed error estimators and prove that the constants in the bounds are independent of the Lam\'{e} coefficients: thus the proposed estimators are robust in the incompressible limit. Numerical results are presented that validate the theoretical estimates. The software used to generate these results is available online.