We consider the nearly incompressible linear elasticity problem with an uncertain spatially varying Young's modulus. The uncertainty is modeled 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. Eigenvalue bounds for the preconditioned system are established and shown to be independent of the discretization parameters and the Poisson ratio. The S-IFISS software used for computation is available online.
- Linear elasticity
- Mixed approximation
- Stochastic galerkin finite element method
- Uncertain material parameters