  In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authors
proposed a new practical method for computing expected values of functionals of
solutions for certain classes of elliptic partial differential equations with
random coefficients. This method was based on combining quasi-Monte Carlo (QMC)
methods for computing the expected values with circulant embedding methods for
sampling the random field on a regular grid. It was found capable of handling
fluid flow problems in random heterogeneous media with high stochastic
dimension, but a convergence theory was missing. This paper provides a
convergence analysis for the method in the case when the QMC method is a
specially designed randomly shifted lattice rule. The convergence result
depends on the eigenvalues of the underlying nested block circulant matrix and
can be independent of the number of stochastic variables under certain
assumptions. In fact the QMC analysis applies to general factorisations of the
covariance matrix to sample the random field. The error analysis for the
underlying fully discrete finite element method allows for locally refined
meshes (via interpolation from a regular sampling grid of the random field).
Numerical results on a non-regular domain with corner singularities in two
spatial dimensions and on a regular domain in three spatial dimensions are
included.
