Robust error estimation for lowest-order approximation of nearly incompressible elasticity

We consider so-called Herrmann and Hydrostatic mixed formulations of classical linear elasticity and analyse the error associated with locally stabilised $P_1-P_0$ finite element approximation. First, we prove a stability estimate for the discrete problem and establish an a priori estimate for the associated energy error. Second, we consider a residual-based a posteriori error estimator as well as a local Poisson problem estimator. We establish bounds for the energy error that are independent of the Lam\'{e} coefficients and prove that the estimators are robust in the incompressible limit. A key issue to be addressed is the requirement for pressure stabilisation. Numerical results are presented that validate the theory. The software used is available online.