An Optimal Solver for Linear Systems Arising from Stochastic FEM Approximation of Diffusion Equations with Random Coefficients

This paper discusses the design and implementation of efficient solution algorithms for symmetric linear systems associated with stochastic Galerkin approximation of elliptic PDE problems with correlated random data. The novel feature of our preconditioned MINRES solver is the incorporation of error control in the natural “energy” norm in combination with a reliable and efficient a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. The MATLAB codes used in the numerical studies are available online.