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We consider the diffusion equation in the setting of operator theory. Inparticular, we study the characterization of the limit of the diffusionoperator for diffusivities approaching zero on a subdomain $\Omega_1$ of thedomain of integration of $\Omega$. We generalize Lions' results to covering thecase of diffusivities which are piecewise $C^1$ up to the boundary of$\Omega_1$ and $\Omega_2$, where $\Omega_2 := \Omega \setminus\overline{\Omega}_1$ instead of piecewise constant coefficients. In addition,we extend both Lions' and our previous results by providing the strongconvergence of $(A_{\bar{p}_\nu}^{-1})_{\nu \in \mathbb{N}^\ast},$ for amonotonically decreasing sequence of diffusivities $(\bar{p}_\nu )_{\nu \in\mathbb{N}^\ast}$.

On the characterization of asymptotic cases of the diffusion equation with rough coefficients and applications to preconditioning