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arXiv · 0902.0788

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

Burak Aksoylu·Horst R. Beyer
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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 $Ω_1$ of the domain of integration of $Ω$. We generalize Lions' results to covering the case of diffusivities which are piecewise $C^1$ up to the boundary of $Ω_1$ and $Ω_2$, where $Ω_2 := Ω\setminus \overlineΩ_1$ instead of piecewise constant coefficients. In addition, we extend both Lions' and our previous results by providing the strong convergence of $(A_{\bar{p}_ν}^{-1})_{ν\in \mathbb{N}^\ast},$ for a monotonically decreasing sequence of diffusivities $(\bar{p}_ν)_{ν\in \mathbb{N}^\ast}$.

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Burak Aksoylu, Horst R. Beyer. 2009-02-04. On the characterization of asymptotic cases of the diffusion equation with rough coefficients and applications to preconditioning. https://arxiv.org/abs/0902.0788

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