  We introduce a new iterative method for computing solutions of elliptic
equations with random rapidly oscillating coefficients. Similarly to a
multigrid method, each step of the iteration involves different computations
meant to address different length scales. However, we use here the homogenized
equation on all scales larger than a fixed multiple of the scale of oscillation
of the coefficients. While the performance of standard multigrid methods
degrades rapidly under the regime of large scale separation that we consider
here, we show an explicit estimate on the contraction factor of our method
which is independent of the size of the domain. We also present numerical
experiments which confirm the effectiveness of the method, with openly
available source code.
