In Chapter 3 we establish the O(√ε) error estimates for some two-scale expansions in H^{1} and the O(ε) convergence rate for solutions u_{ε} in L^{2}. The results are obtained without any smoothness assumption on the coefficient matrix A. In this chapter we return to the problem of convergence rates and prove various results under some additional smoothness assumptions, using uniform regularity estimates obtained in Chapters 4–6. We shall be mainly interested in the sharp O(ε) or near sharp rates of convergence.