Interior error estimate for periodic homogenization

In a previous article about the homogenization of the classical problem of diff usion in a bounded domain with su ciently smooth boundary we proved that the error is of order $\epsilon^{1/2}$. Now, for an open set with su ciently smooth boundary $C^{1,1}$ and homogeneous Dirichlet or Neuman limits conditions we show that in any open set strongly included in the error is of order $\epsilon$. If the open set $\Omega\subset R^n$ is of polygonal (n=2) or polyhedral (n=3) boundary we also give the global and interrior error estimates.