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

Localization phenomena (sometimes called “flea on the elephantu0027u0027) for the operator $L^varepsilon=-varepsilon^2 Delta u + p(mathbf{x}) u$, $p(mathbf{x})$ 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--Poincare operator for $L^varepsilon$, and for which error estimates are established. Such a two-dimensional discretization produces less mesh imprinting than more standard finite differences and correctly captures sharp layers.