 Two-scale expansion for periodic conductivity

A periodic expansion with $y=x/ϵ$ separates the macroscopic variable from the cell variable. Its leading cell equation forces $u_0$ to be independent of $y$, the next equation gives $u_1=w_j(y){\partial }_{x_j}{u}_0$, and averaging the order-one flux equation yields the homogenized divergence equation. Bulk correctors generally require boundary-layer adjustments at a physical Dirichlet boundary.