A direct boundary integral equation method for transmission problems

Cloaking of Small Objects by Anomalous Localized Resonance

“…Indeed, with a value −κ in the cloak, κ = 1, the effect of anomalous resonance disappears as observed in Subsection 2.1. This is consistent with the results in [1,2,6] (see also the references therein), where well posedness of a diffraction problem is shown under the condition that |κ − 1| is large enough.…”

“…This problem can not be treated by standard methods, for elaborate approaches we refer to [1,2,6]. Another approach is to consider the case of a positive dissipation coefficient η and to study the limit η → 0, which might be considered also a low viscosity limit [16].…”

Pointwise convergence of some boundary element methods. Part II

“…Operators of positive order also come up in boundary element Galerkin methods defined via hypersingular intégral operators ; see [7], [8], [9], [10], [12], [13], [14], [21], [22], [37], As an example and model operator let us consider the normal derivative of the double layer potential which is associated to Neumann problems, see [2], [7], [21], [38], ( Here, y n dénotes the fundamental solution of the Laplacian in 0? ", ft *\ n = 2 or n = 3, -and -are the normal derivatives with respect to the dn x dn y exterior unit normal to F at x and y, respectively, and K(x,y) is a sufficiently smooth kernel.…”

“…In acoustics [21], elasticity [22], electromagnetics [8] and flow problems [14] one finds hypersingular strongly elliptic boundary intégral équations of order 2a = 1. Hère (4.7) provides the pointwise quadratic order convergence of the second derivatives and almost cubic order pointwise convergence of the first derivatives and the function values with piecewise cubic spline collocation.…”

Approximation by generalized impedance boundary conditions of a transmission problem in acoustic scattering

“…To numerically solve such a problem, one may solve a system of coupled equivalent integral equations [8,16] set on Γ. This provides an accurate solution [9,18,24]. However, the handling of two distinct domains defined by some different constitutive parameters involves two wavelengths: λ 1 and λ 2 .…”