 Periodic homogenization of a diffusion equation

Consider $u_t={\partial }_x[a(x/\epsilon )u_x]$, where $a$ has period one and $0<a_{-}\le a(y)\le a_{+}<\infty$. At scales large compared with $\epsilon$, the one-dimensional effective diffusion coefficient isThe harmonic mean comes from flux continuity. Impose a unit mean gradient across one microscopic cell and let $v^{\prime }(y)$ be its local gradient. A constant conservative flux requires $a(y)v^{\prime }(y)=q$. The unit total change gives $1={\int }_0^1{v}^{\prime }=q{\int }_0^1{a}^{-1}$, so $q=a_{\mathrm{eff}}$. The associated cell correction is periodic because $v(y)-y$ has zero change across a period.

For two equally thick layers with coefficients $a_1,a_2>0$, this yields $a_{\mathrm{eff}}=2a_1{a}_2/(a_1+a_2)$. Taking $a_1=1$ and $a_2=9$ gives $1.8$, rather than the arithmetic mean $5$. The less diffusive layer requires a larger local gradient to carry the same flux. This explicit harmonic-mean formula is one-dimensional; a multidimensional periodic medium generally has an effective diffusion tensor.