Periodic homogenization of a diffusion equation

For $u_t={\partial }_x[a(x/ϵ)u_x]$ with positive periodic $a$, the one-dimensional homogenized diffusivity is the harmonic mean $a_{\mathrm{eff}}=⟨a^{-1}{⟩}^{-1}$. Microscopic flux continuity produces this harmonic rather than arithmetic averaging.