Solving log-transformed random diffusion problems by stochastic Galerkin mixed finite element methods

Stochastic Galerkin finite element discretisations of PDEs with stochastically nonlinear coefficients lead to linear systems of equations with block dense matrices. In contrast, stochastic Galerkin finite element discretisations of PDEs with stochastically linear coefficients lead to linear systems of equations with block sparse matrices which are cheaper to manipulate and precondition in the framework of Krylov subspace iteration. In this paper we focus on mixed formulations of second-order elliptic problems, where the diffusion coefficient is the exponential of a random field, and the priority is to approximate the flux. We build on the previous work [Efficient iterative solvers for stochastic Galerkin discretizations of log-transformed random diffusion problems, SIAM J. Sci. Comput., 34(2012), pp.A659–A682] and reformulate the PDE model as a first-order system in which the logarithm of the diffusion coefficient appears on the left-hand side. We apply a stochastic Galerkin mixed finite element method and discuss block triangular and block diagonal preconditioners for use with GMRES iteration. In particular, we analyse a practical approximation to the Schur complement of the Galerkin matrix and provide spectral inclusion bounds. Numerical experiments reveal that the preconditioners are completely insensitive to the spatial mesh size, and are only slightly sensitive to the statistical parameters of the diffusion coefficient. As a result, the computational cost required to approximate the flux when the diffusion coefficient is stochastically nonlinear grows only linearly with respect to the total problem size.