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

2009/02/04 by Burak Aksoylu, Aksoylu, Burak, Horst R. Beyer +1
Mathematics · #35J25 #47F05 #65J10 #65N99 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.FA #msc:35J25 #msc:47F05 #msc:65J10 #msc:65N99

paper · pdf · doi:10.48550/arxiv.0902.0788

arxiv created 2009/02/04 · arxiv updated 2009/12/01

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 Ω₁ of the domain of integration of Ω. We generalize Lions' results to covering the case of diffusivities which are piecewise C¹ up to the boundary of Ω₁ and Ω₂, where Ω₂ := Ω∖ Ω₁ instead of piecewise constant coefficients. In addition, we extend both Lions' and our previous results by providing the strong convergence of (A_p_ν^-1)_{ν∈ ℕ^∗}, for a monotonically decreasing sequence of diffusivities (p_ν)_{ν∈ ℕ^∗}.

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