In the last few years, there has been renewed interest in stochastic
finite element methods (SFEMs), which facilitate the approximation
of statistics of solutions to PDEs with random data. SFEMs based on
sampling, such as stochastic collocation schemes, lead to decoupled
problems requiring only deterministic solvers. SFEMs based on
Galerkin approximation satisfy an optimality condition but require
the solution of a single linear system of equations that couples
deterministic and stochastic degrees of freedom. This is regarded as
a serious bottleneck in computations and the difficulty is even more
pronounced when we attempt to solve systems of PDEs with
random data via stochastic mixed FEMs.
In this talk, we give an overview of solution strategies for the
saddle-point systems that arise when the mixed form of the Darcy
flow problem, with correlated random coefficients, is discretised
via stochastic Galerkin and stochastic collocation techniques. For
the stochastic Galerkin approach, the systems are orders of
magnitude larger than those arising for deterministic problems. We
report on fast solvers and preconditioners based on multigrid, which
have proved successful for deterministic problems. In particular, we
examine their robustness with respect to the random diffusion
coefficients, which can be either a linear or non-linear function of
a finite set of random parameters with a prescribed probability
distribution.