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

We consider the diffusion equation in the setting of operator theory. In particular, we study the characterization of the limit of the diffusion operator for diffusivities approaching zero on a subdomain $\Omega_1$ of the domain of integration of $\Omega$. We generalize Lions' results to covering the case 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 strong convergence of $(A_{\bar{p}_\nu}^{-1})_{\nu \in \mathbb{N}^\ast},$ for a monotonically decreasing sequence of diffusivities $(\bar{p}_\nu )_{\nu \in \mathbb{N}^\ast}$.