Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field

François Alouges (1) , Virginie Bonnaillie-Noël (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IRMAR - Institut de Recherche Mathématique de Rennes

Motivated by questions arising in the theory of superconductivity, we are interested in the study of the fundamental state of the Schrodinger operator with magnetic field in a domain with corners. Although this problem has been extensively studied theoretically, many fewer articles deal with numerical approaches. In this article, we propose numerical experiments based on the finite element method to determine the bottom of the spectrum of the operator. Analyzing the drawbacks of a standard method and the properties of the operator, we propose a natural gauge-invariant method and provide a few numerical simulations. We furthermore improve the numerical results by coupling the method with a mesh-refinement technique based on a posteriori error estimates developed by one of the authors. This allows us to look at the monotonicity of the smallest eigenvalue in an angular sector with respect to the angle, which complement theoretical studies

Domaines: Analyse numérique [math.NA]

François Alouges, Virginie Bonnaillie-Noël. Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field. Numerical Methods for Partial Differential Equations, 2006, 22 (5), pp.1090-1105. ⟨10.1002/num.20137⟩. ⟨hal-00447135⟩