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

TL;DR

This paper analyzes the asymptotic behavior of the diffusion operator with rough, piecewise smooth coefficients, extending classical results and demonstrating strong convergence of inverse operators for decreasing diffusivities, with applications to preconditioning.

Contribution

It generalizes Lions' results to piecewise $C^1$ diffusivities and proves strong convergence of inverse operators for decreasing diffusivities.

Findings

Extended characterization of diffusion operators with piecewise smooth coefficients.

Proved strong convergence of inverse operators for monotonic diffusivity sequences.

Applicable to preconditioning in operator theory.

Abstract

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 $(\bar{p}_\nu )_{\nu \in \mathbb{N}^\ast}$.