具有粗糙系数的扩散方程渐近情形的特征及其在预处理中的应用
On the characterization of asymptotic cases of the diffusion equation with rough coefficients and applications to preconditioning
摘要：
在算子理论的背景下考虑扩散方程。特别是，我们研究的极限的扩散算子的扩散系数接近零的子域$Ω_1$的积分域集成$Ω$的表征。我们将Lions的结果推广到扩散系数为分段$C^1$直到$Ω_1$和$Ω_2$的边界的情形，其中用$Ω_2 := Ω\setminus \overlineΩ_1$代替分段常系数.此外，我们通过为扩散率单调递减序列$(\bar{p}_ν)_{ν\in \mathbb{N}^\ast}$提供$(A_{\bar{p}_ν}^{-1})_{ν\in \mathbb{N}^\ast},$的强收敛性，扩展了Lions和我们之前的结果。
英文摘要：
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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