  The focus of this work is a posteriori error estimation for stochastic
Galerkin approximations of parameter-dependent linear elasticity equations. The
starting point is a three-field PDE model in which the Young's modulus is an
affine function of a countable set of parameters. We analyse the weak
formulation, its stability with respect to a weighted norm and discuss
approximation using stochastic Galerkin mixed finite element methods
(SG-MFEMs). We introduce a novel a posteriori error estimation scheme and
establish upper and lower bounds for the SG-MFEM error. The constants in the
bounds are independent of the Poisson ratio as well as the SG-MFEM
discretisation parameters. In addition, we discuss proxies for the error
reduction associated with certain enrichments of the SG-MFEM spaces and we use
these to develop an adaptive algorithm that terminates when the estimated error
falls below a user-prescribed tolerance. We prove that both the a posteriori
error estimate and the error reduction proxies are reliable and efficient in
the incompressible limit case. Numerical results are presented to validate the
theory. All experiments were performed using open source (IFISS) software that
is available online.
