### Abstract

In this paper we describe a fast parallel method for highly
ill-conditioned saddle-point systems arising from mixed finite element
simulations of stochastic partial differential equations (PDEs) modelling flow
in heterogeneous media. Each realisation of these stochastic PDEs requires
the solution of the linear first-order velocity-pressure
system comprising Darcy's law coupled with an
incompressibility constraint. The chief difficulty is that
the permeability may be highly variable, especially when the statistical model
has a large variance and a small correlation length.
For reasonable accuracy, the discretisation has to be extremely fine.
We solve these problems by first reducing the saddle-point formulation to a
symmetric positive definite (SPD) problem using a suitable basis for the
space of divergence-free velocities.
The reduced problem is solved using parallel conjugate gradients
preconditioned with an algebraically determined additive Schwarz
domain decomposition preconditioner. The result is a solver which
exhibits a good degree of robustness with respect to
the mesh size as well as to the variance and to physically relevant
values of the correlation length of the underlying
permeability field. Numerical experiments exhibit
almost optimal levels of parallel efficiency.
The domain decomposition solver
(DOUG)
used here is applicable not only to this problem but can be used to solve
general unstructured finite element systems on a wide range of parallel
architectures.