A numerical scheme was presented for the analysis of steady-state non-linear diffusion in random heterogeneous media. The key was to solve the non-linear stochastic governing equations iteratively via an inexact Picard iteration scheme, in which the non-linear constitutive law was linearized by using the current estimate of the solution. The linearized stochastic governing equations were then spatially discretized and solved approximately by using stochastic reduced basis projection schemes. The approximation to the solution thus obtained was used as an estimate for the next iteration. This iterative procedure was repeated until an appropriate convergence criterion was met. Detailed numerical studies were presented for diffusion within a square domain, for various degrees of non-linearity. The numerical results were compared with benchmark Monte Carlo simulations, and it was shown that the proposed approach provided good approximations to the response statistics for modest computational effort.

Inexact Picard Iterative Scheme for Steady-State Nonlinear Diffusion in Random Heterogeneous Media. P.S.Mohan, P.B.Nair, A.J.Keane: Physical Review E, 2009, 79[4], 046706