Catherine Powell (U. Manchester): Adaptive & Multilevel Stochastic Galerkin FEM Approximation for PDEs with Random Inputs

In this talk, we give an overview of some recent work on the development of adaptive and multilevel stochastic Galerkin finite element methods (SGFEMs) for solving parameter-dependent PDEs (or PDEs with uncertain inputs), focusing on a posteriori error estimation and linear algebra issues. The main ideas underpinning the error estimation strategy will be outlined first for a simple scalar diffusion problem. Unlike standard residual-based error estimation schemes, the proposed strategy requires the solution of auxiliary problems on carefully constructed detail spaces on both the spatial and parameter domains. We establish two-sided bounds for the SGFEM approximation error in terms of the proposed estimator. In addition, we obtain estimates of the error reduction associated with potential enrichments of the SGFEM space and suggest how to use these to develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. The extension of this methodology to more complex systems of PDEs, such as locking-free mixed formulations of the linear elasticity equations, will also briefly be discussed.