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

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 body is almost incompressible. Several alternative a posteriori error estimators for the energy norm of the finite element error are introduced and analysed. We establish upper and lower bounds for the energy error in terms of these error estimators and show that the constants in the bounds are independent of the Lamé coefficients: the proposed estimators are robust in the incompressible limit. We also consider the requirement for pressure stabilization when using lowest-order conforming approximation. Computational results are presented that validate the theoretical estimates. The software used to generate these results is available online.