# On the characterization of asymptotic cases of the diffusion equation with rough coefficients and applications to preconditioning - Mathematics > Analysis of PDEs

On the characterization of asymptotic cases of the diffusion equation with rough coefficients and applications to preconditioning - Mathematics > Analysis of PDEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: 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 $A {\bar{p} u}^{-1} { u \in \mathbb{N}^\ast},$ for amonotonically decreasing sequence of diffusivities $\bar{p} u { u \in\mathbb{N}^\ast}$.

Author: ** Burak Aksoylu, Horst R. Beyer**

Source: https://arxiv.org/