Approximations of the resolvent for a non-self-adjoint diffusion operator with rapidly oscillating coefficients

A strongly inhomogeneous diffusion operatorwith drift depending on a small parameter ? is studied in the space L ²(? ⁿ ). The strong inhomogeneity consists in that the coefficients of the operator are ?-periodic and, in addition, the drift vector is of the order of ? ^?1. As ? → 0, approximations in the operator L ²-norm of order ? and ? ² are constructed for the resolvent of the operator. For each of these orders of approximation, an averaged diffusion operator is obtained. A spectral method based on the Bloch representation for an operator with periodic coefficients is used.