  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.
