Numerical solution of the homogeneous Neumann boundary value problem on domains with a thin layer of random thickness

The present article is dedicated to the numerical solution of homogeneous Neumann boundary value problems on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on the random domain can be transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Ventcell boundary condition and yields a second order accurate solution in the scale parameter  ε of the layer's thickness. With the help of the Karhunen-Loeve expansion, we transform this random boundary value problem into a deterministic, parametric one with a possibly high-dimensional parameter y. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter y which are robust in the scale parameter  ε . Numerical results validate our theoretical findings

Two-parameter asymptotic expansions for elliptic equations with small geometric perturbation and high contrast ratio

We consider the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirich- let boundary condition and transmission condition, subject to the small geometric perturbation and the high contrast ratio of the conductivity. We consider two types of perturbations: the first corresponds to a thin layer coating a fixed bounded domain and the second is the per perturbation of the interface. As the perturbation size tends to zero and the ratio of the conductivities in two subdomains tends to zero, the two-parameter asymptotic expansions on the fixed reference domain are derived to any order after the single parameter expansions are solved be- forehand. Our main tool is the asymptotic analysis based on the Taylor expansions for the properly extended solutions on fixed domains. The Neumann boundary condition and Robin boundary condition arise in two-parameter expansions, depending on the relation of the geometric perturbation size and the contrast ratio

Cloaking and anamorphism for light and mass diffusion

We first review classical results on cloaking and mirage effects for electromagnetic waves. We then show that transformation optics allows the masking of objects or produces mirages in diffusive regimes. In order to achieve this, we consider the equation for diffusive photon density in transformed coordinates, which is valid for diffusive light in scattering media. More precisely, generalizing transformations for star domains introduced in [Diatta and Guenneau, J. Opt. 13, 024012, 2011] for matter waves, we numerically demonstrate that infinite conducting objects of different shapes scatter diffusive light in exactly the same way. We also propose a design of external light-diffusion cloak with spatially varying sign-shifting parameters that hides a finite size scatterer outside the cloak. We next analyse non-physical parameter in the transformed Fick's equation derived in [Guenneau and Puvirajesinghe, R. Soc. Interface 10, 20130106, 2013], and propose to use a non-linear transform that overcomes this problem. We finally investigate other form invariant transformed diffusion-like equations in the time domain, and touch upon conformal mappings and non-Euclidean cloaking applied to diffusion processes.Comment: 42 pages, Latex, 14 figures. V2: Major changes : some formulas corrected, some extra cases added, overall length extended from 21 pages (V1) to 42 pages (present version V2). The last version will appear at Journal of Optic