Bert Seynaeve : A multigrid approach to the stachastic finite element method for PDEs with stachastic parameters
We consider the numerical solution of elliptic or parabolic partial differential
equations with stochastic coefficients. Such equations appear in reliability problems.
Various approaches exist for dealing with the uncertainty propagation question: Monte Carlo methods,
perturbation techniques, variance propagation, etc. Here, we deal with the stochastic
finite element method (SFEM) . This method transforms a system of PDEs with stochastic parameters
into a stochastic linear system Ku = f, by means of a finite element Galerkin discretization.
The finite element solution vector u, with n nodal stochastic unknowns, is approximated by a
linear combination of N deterministic vectors. The coefficients are orthogonal polynomials in
the random variables (that occur in K and f) . The solution u can be found as the solution of
an very large deterministic linear system with n x N unknowns. Unlike commonly used methods such as
the perturbation method, 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.
We will show that discretized PDEs with stochastic coefficients can also be solved by using a
linear iterative method (e.g. Gauss-Seidel), adapted to the stochastic case by replacing every equation
by a relatively small linear system (of size N). The computational efficiency can then be improved
by using that method as a smoother within a multigrid context. The solution obtained by our method
is given by a stochastic expansion of the same type as the one used in the classical SFEM-approach.
The coefficients can differ slightly, however, depending on the choice of the iterative method.
The new approach also allows every element in the solution vector to be a rational expression in
the random variables instead of a polynomial expression, which would be computationally prohibitive
with the classical approach.
 R.G. Ghanem and P.D. Spanos. Stochastic finite elements: a spectral approach. Springer-Verlag, New Yo