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In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authorsproposed a new practical method for computing expected values of functionals ofsolutions for certain classes of elliptic partial differential equations withrandom coefficients. This method was based on combining quasi-Monte Carlo (QMC)methods for computing the expected values with circulant embedding methods forsampling the random field on a regular grid. It was found capable of handlingfluid flow problems in random heterogeneous media with high stochasticdimension, but a convergence theory was missing. This paper provides aconvergence analysis for the method in the case when the QMC method is aspecially designed randomly shifted lattice rule. The convergence resultdepends on the eigenvalues of the underlying nested block circulant matrix andcan be independent of the number of stochastic variables under certainassumptions. In fact the QMC analysis applies to general factorisations of thecovariance matrix to sample the random field. The error analysis for theunderlying fully discrete finite element method allows for locally refinedmeshes (via interpolation from a regular sampling grid of the random field).Numerical results on a non-regular domain with corner singularities in twospatial dimensions and on a regular domain in three spatial dimensions areincluded.

Circulant embedding with QMC -- analysis for elliptic PDE with lognormal coefficients