  We consider a class of multiscale parabolic problems with diffusion
coefficients oscillating in space at a possibly small scale $\varepsilon$.
Numerical homogenization methods are popular for such problems, because they
capture efficiently the asymptotic behaviour as $\varepsilon \rightarrow 0$,
without using a dramatically fine spatial discretization at the scale of the
fast oscillations. However, known such homogenization schemes are in general
not accurate for both the highly oscillatory regime $\varepsilon \rightarrow 0$
and the non oscillatory regime $\varepsilon \sim 1$. In this paper, we
introduce an Asymptotic Preserving method based on an exact micro-macro
decomposition of the solution which remains consistent for both regimes.
