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In this paper, we propose and analyze a new stochastic homogenization methodfor diffusion equations with random and fast oscillatory coefficients. In theproposed method, the homogenized solutions are sought through a two-stageprocedure. In the first stage, the original oscillatory diffusion equation isapproximated, for each fixed random sample w, by a spatially homogenizeddiffusion equation with piecewise constant coefficients, resulting a randomdiffusion equation. In the second stage, the resulted random diffusion equationis approximated and computed by using an efficient multi-modes Monte Carlomethod which only requires to solve a diffusion equation with a constantdiffusion coefficient and a random right-hand side. The main advantage of theproposed method is that it separates the computational difficulty caused by thespatial fast oscillation of the solution and that caused by the randomness ofthe solution, so they can be overcome separately using different strategies.The convergence of the solution of the spatially homogenized equation (from thefirst stage) to the solution of the original random diffusion equation isestablished and the optimal rate of convergence is also obtained for theproposed multi-modes Monte Carlo method. Numerical experiments on somebenchmark test problems for random composite materials are also presented togauge the efficiency and accuracy of the proposed two-stage stochastichomogenization method.

An efficient multi-modes Monte Carlo homogenization method for random materials