Robust preconditioning for stochastic Galerkin formulations of parameter-dependent nearly incompressible linear elasticity equations

TL;DR

This paper develops a robust preconditioning strategy for stochastic Galerkin methods applied to nearly incompressible linear elasticity problems with uncertain material properties, ensuring efficient and stable solutions.

Contribution

A novel three-field mixed variational formulation and a preconditioner for the resulting linear systems that are independent of discretisation and material parameters.

Findings

Eigenvalue bounds are independent of discretisation parameters.

The preconditioner improves the efficiency of the MINRES solver.

The approach is validated using the S-IFISS software.

Abstract

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.