A Two-Dimensional “Flea on the Elephant” Phenomenon and its Numerical Visualization

Abstract

Localization phenomena (sometimes called "flea on the elephant") for the operator L ε = −ε 2 ∆u + p(x)u, p(x) being an asymmetric double-well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincaré operator for L ε , and for which error estimates are established. Such a two-dimensional discretization produces less mesh-imprinting than more standard finite-differences and captures correctly sharp layers.

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

“…This was actually the case initially considered in [15]. • More recently a numerical analysis of the effect is proposed in [2].…”

“…Formally, the main term of the perturbation is indeed δ w(x)u (x) 2 dx , and we use a lower bound for the decay of u on the support of w. This implies the existence of δ 1 > 0 such that, for 0 < δ < δ 1 , we have as h → 0,…”

“…3. Better than that, we put into evidence Simon's "flea on the elephant phenomenon" for the localization of the Steklov eigenfunctions on the boundary (see [2,13,15,20]) and Appendix B). Precisely, we prove that for symmetric (with respect to 1 2 ) warped products, the Steklov eigenfunctions concentrate at both connected components of the boundary as m → ∞.…”

Exponential localization of Steklov eigenfunctions on warped product manifolds: the flea on the elephant phenomenon

“…3. Better than that, we put into evidence Simon's "flea on the elephant phenomenon" for the localization of the Steklov eigenfunctions on the boundary (see [2,13,15,21]) and appendix B). Precisely, we prove that for symmetric (with respect to 1 2 ) warped products, the Steklov eigenfunctions concentrate at both connected components of the boundary as m → ∞.…”

Diffusive limits of 2D well-balanced schemes for kinetic models of neutron transport

“…The scheme (31) works on a 9-points "Moore stencil", hence doesn't match the "Steklov scheme", see [20, §4] and [4], which has only a 5-points stencil. Points on the diagonals ρ n i±1,j±1 are multiplied by…”