Computations of the first eigenpairs for the Schrödinger operator with magnetic field

Virginie Bonnaillie-Noël, Monique Dauge, Daniel Martin, Gregory Vial

This paper is devoted to computations of eigenvalues and eigenvectors for the Schrödinger operator with constant magnetic field in a domain with corners, as the semi-classical parameter   h  tends to   0. The eigenvectors corresponding to the smallest eigenvalues concentrate in the corners: They have a two-scale structure, consisting of a corner layer at scale   \sqrt(h) and an oscillatory term at scale   h . The high frequency oscillations make the numerical computations particularly delicate. We propose a high order finite element method to overcome this difficulty. Relying on such a discretization, we illustrate theoretical results on plane sectors, squares, and other straight or curved polygons. We conclude by discussing convergence issues.

12 December 2005. Preprint IRMAR 05-37.

Computer Methods in Applied Mechanics and Engineering, 196 (2007) 3841-3858.

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