Robust preconditioning for stochastic Galerkin formulations of parameter-dependent nearly incompressible linear elasticity equations
It addresses the computational challenge of solving stochastic Galerkin systems for nearly incompressible elasticity, a problem relevant to uncertainty quantification in solid mechanics.
The paper introduces a three-field mixed variational formulation for nearly incompressible linear elasticity with uncertain Young's modulus, and develops a preconditioner for the resulting linear system that yields eigenvalue bounds independent of discretization parameters and Poisson ratio.
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.