STOCHASTIC GALERKIN DISCRETIZATION OF THE LOG-NORMAL ISOTROPIC DIFFUSION PROBLEM

The stochastic Galerkin method is developed for the isotropic diffusion equation with an unbounded random diffusion coefficient. The logarithm of the diffusion coefficient is assumed to be an infinite series of Gaussian random variables. Well-posed weak formulations of the model problem are derived on standard Bochner–Lebesgue spaces. The Galerkin solution is shown to be almost quasi-optimal in the sense that the error of the Galerkin projection can be estimated by a best approximation error in a slightly stronger norm. As a result, convergence analysis of the stochastic Galerkin method is reduced to the problem of approximating the Hermite coefficients of the exact solution by standard finite elements.