  We design a numerical scheme for transport equations with oscillatory
periodic scattering coefficients. The scheme is asymptotic preserving in the
diffusion limit as Knudsen number goes to zero. It also captures the
homogenization limit as the length scale of the scattering coefficient goes to
zero. The proposed method is based on the construction of multiscale finite
element basis and a Galerkin projection based on the even-odd decomposition.
The method is analyzed in the asymptotic regime, as well as validated
numerically.
