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We describe the numerical scheme for the discretization and solution of 2Delliptic equations with strongly varying piecewise constant coefficientsarising in the stochastic homogenization of multiscale composite materials. Anefficient stiffness matrix generation scheme based on assembling the localKronecker product matrices is introduced. The resulting large linear systems ofequations are solved by the preconditioned CG iteration with a convergence ratethat is independent of the grid size and the variation in jumping coefficients(contrast). Using this solver we numerically investigate the convergence of theRepresentative Volume Element (RVE) method in stochastic homogenization thatextracts the effective behavior of the random coefficient field. Our numericalexperiments confirm the asymptotic convergence rate of systematic error andstandard deviation in the size of RVE rigorously established in [6]. Theasymptotic behavior of covariances of the homogenized matrix in the form of aquartic tensor is also studied numerically. Our approach allows laptopcomputation of sufficiently large number of stochastic realizations even forlarge sizes of the RVE.

Numerical study in stochastic homogenization for elliptic PDEs: convergence rate in the size of representative volume elements