We consider the numerical solution of elliptic and parabolic partial differential equations with stochastic coefficients. Such equations appear e.g. in reliability problems. Various approaches exist for dealing with such 'uncertainty propagation' models: Monte Carlo methods, perturbation techniques, variance propagation, etc. Here, we deal with the stochastic finite element method (SFEM) [1]. This method transforms a system of PDEs with stochastic parameters into a stochastic linear system by means of a finite element Galerkin discretization. The stochastic solution vector of this system is approximated by a linear combination of deterministic vectors; the coefficients are orthogonal polynomials in the random variables. Unlike commonly used methods such as the perturbation method, the SFEM gives a result that contains all stochastic characteristics of the solution. It also improves Monte Carlo methods significantly because sampling can be done after solving the system of PDEs.

In order to solve the discretized stochastic system that appears in the SFEM, stochastic versions of iterative methods can be applied, and their convergence can be accelerated by implementing them in a multigrid context [2]. In the work we present here, the convergence properties of these stochastic iterative methods and multigrid methods are investigated theoretically: deterministic (local) Fourier-mode analysis techniques are extended to the stochastic case by taking the eigenstructure into account of a matrix that depends on the random structure of the problem. This is equivalent to choosing an alternative set of polynomial basis functions in the random variables, which results in a decoupling of the original stochastic problem into a very large number of deterministic problems of the same type. The theoretical convergence rates that we obtain predict the results of numerical experiments very well, and the results of the analysis can also be used to design optimal stochastic multigrid algorithms.

References:
[1] R.G. Ghanem and P.D. Spanos. 'Stochastic finite elements: a spectral approach'. Springer-Verlag, New York, 1991.
[2] O.P. Le Maître et.al. 'A multigrid solver for two-dimensional stochastic diffusion equations', Computer Methods in Applied Mechanics and Engineering, Vol. 192, Iss. 41-42, 2003, pp. 4723-4744.