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We derive the optimal energy error estimate for multiscale finite elementmethod with oversampling technique applying to elliptic system with rapidlyoscillating periodic coefficients under the assumption that the coefficientsare bounded and measurable, which may admit rough microstructures. As aby-product of the energy estimate, we derive the rate of convergence inL$^{d/(d-1)}-$norm.

Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited