Analyse mathématique de la supraconductivité dans un domaine à coins: méthodes semi-classiques et numériques

Abstract : Superconducting theory, modelized by Ginzburg and Landau, motivates works about Schrödinger operator with magnetic field. The aim of this thesis is to analyze the geometry influence on the superconductivity emergence by extending results for regular domains to domains with corners. Semi-classical analysis leads to deal with three model operators: the Neumann realization of the Schrödinger operator with constant magnetic field in the plane, the half-plane and angular sectors. The study of the two first is well known and we concentrate on the third. After determination of the bottom of the essential spectrum, we show that the bottom of the spectrum is an eigenvalue for a sector with an acute angle. We give an explicit asymptotics of the lowest eigenvalue as the angle tends to 0 and clarify the localization of the ground state using Agmon's techniques. Then we illustrate and estimate the behavior of eigenvectors and eigenvalues thanks to numerical tools based on finite elements method. Due to the ground state localization, the discretized problem is badly conditioned; nevertheless the analysis of the operator properties and of the drawbacks of classical methods lead us to implement a robust and efficient algorithm computing the ground state. To improve numerical results, we construct a posteriori error estimates for this eigenvalue problem and use the mesh-refinement techniques to localize the eigenstate associated to general domains and to study the variation of the bottom of the spectrum according to the angle of the sector.
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https://tel.archives-ouvertes.fr/tel-00005430
Contributor : Virginie Bonnaillie <>
Submitted on : Tuesday, March 23, 2004 - 11:10:33 AM
Last modification on : Thursday, April 11, 2019 - 1:11:18 AM
Long-term archiving on : Friday, April 2, 2010 - 8:05:38 PM

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Virginie Bonnaillie. Analyse mathématique de la supraconductivité dans un domaine à coins: méthodes semi-classiques et numériques. Mathématiques [math]. Université Paris Sud - Paris XI, 2003. Français. ⟨tel-00005430⟩

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