  A stress equilibration procedure for linear elasticity is proposed and
analyzed in this paper with emphasis on the behavior for (nearly)
incompressible materials. Based on the displacement-pressure approximation
computed with a stable finite element pair, it constructs an $H
(\text{div})$-conforming, weakly symmetric stress reconstruction. Our focus is
on the Taylor-Hood combination of continuous finite element spaces of
polynomial degrees $k+1$ and $k$ for the displacement and the pressure,
respectively. Our construction leads then to reconstructed stresses by
Raviart-Thomas elements of degree $k$ which are weakly symmetric in the sense
that its anti-symmetric part is zero tested against continuous piecewise
polynomial functions of degree $k$. The computation is performed locally on a
set of vertex patches covering the computational domain in the spirit of
equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local
problems need to satisfy consistency conditions associated with all rigid body
modes, in contrast to the case of Poisson's equation where only the constant
modes are involved. The resulting error estimator is shown to constitute a
guaranteed upper bound for the error with a constant that depends only on the
shape regularity of the triangulation. Local efficiency, uniformly in the
incompressible limit, is deduced from the upper bound by the residual error
estimator.
