Standard multigrid methods are not so eective for equations with highly oscillatory coecients. New coarse grid operators based on homogenized operators are introduced to restore the fast convergence rate of multigrid methods. Finite dierence approximations are used for the discretization of the equations. The analysis of convergence is based on homogenization theory. Proofs are presented for a two level method with the homogenized coarse grid operator to solve t w o classes of elliptic equations with Dirichlet boundary conditions.