The focus of this talk is the accurate and efficient numerical solution of parametric elliptic PDEs. In particular, we consider elliptic PDEs with diffusion coefficients represented as affine functions of countably infinitely many parameters. Multilevel stochastic Galerkin finite element methods (SGFEMs) provide polynomial-based approximations with spatial coefficients that reside in potentially different finite element spaces. Well established theoretical results state that for nice enough test problems multilevel SGFEMs can achieve rates of convergence independent of the number of input parameters, thereby breaking the curse of dimensionality. However, achieving this in practice using automated computational algorithms remains challenging. We discuss an adaptive multilevel method that is driven by a classical hierarchical a posteriori error estimation strategy—modified for the parametric PDE setting— and present numerical results.
 Adam J. Crowder, Adaptive and Multilevel Stochastic Galerkin Finite Element Methods, PhD Thesis, University of Manchester, 2020.
 Adam J. Crowder, Catherine E. Powell, and Alex Bespalov. Efficient adaptive multilevel stochastic Galerkin approximation using implicit a posteriori error estimation. SIAM Journal on Scientific Computing, 41(3):A1681–A1705, 2019.