A TWO-DIMENSIONAL "FLEA ON THE ELEPHANT" PHENOMENON AND ITS NUMERICAL VISUALIZATION

Roberta Bianchini (1) , Laurent Gosse (2) , Enrique Zuazua (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"
3 Departamento de Matemáticas [Madrid]

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.

Mots clés: Schrödinger equation; Lipschitz domains; Asymmetric double-well potential

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Roberta Bianchini, Laurent Gosse, Enrique Zuazua. A TWO-DIMENSIONAL "FLEA ON THE ELEPHANT" PHENOMENON AND ITS NUMERICAL VISUALIZATION. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 2019, 17 (1), pp.137-166. ⟨hal-01935261⟩