  We consider the nearly incompressible linear elasticity problem with an
uncertain spatially varying Young's modulus. The uncertainty is modelled with a
finite set of parameters with prescribed probability distribution. We introduce
a novel three-field mixed variational formulation of the PDE model and discuss
its approximation by stochastic Galerkin mixed finite element techniques.
First, we establish the well posedness of the proposed variational formulation
and the associated finite-dimensional approximation. Second, we focus on the
efficient solution of the associated large and indefinite linear system of
equations. A new preconditioner is introduced for use with the minimal residual
method (MINRES). Eigenvalue bounds for the preconditioned system are
established and shown to be independent of the discretisation parameters and
the Poisson ratio. The S-IFISS software used for computation is available
online.
