Bespoke stochastic Galerkin approximation of nearly incompressible elasticity

We discuss the key role that bespoke linear algebra plays in modelling PDEs with random coefficients using stochastic Galerkin approximation methods. As a specific example, we consider nearly incompressible linear elasticity problems 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 and  assess the stability with respect to a weighted norm. The main focus will be  the efficient solution of the associated high-dimensional indefinite linear system of equations. Eigenvalue bounds for the preconditioned system can be  established and shown to be independent of the discretisation parameters and the Poisson ratio.  We also  discuss an associated a posteriori error estimation strategy and assess proxies for the error reduction associated with selected enrichments of the approximation spaces.  We will show by example that these proxies enable the design of efficient  adaptive solution algorithms that terminate when the estimated error falls below a user-prescribed tolerance.

This is joint work with Arbaz Khan and Catherine Powell