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

Motivated by questions arising in the theory of superconductivity, we are interested in the study of the fundamental state of the Schrödinger operator with magnetic field in a domain with corners. Although this problem has been extensively studied theoretically, many fewer articles deal with numeric...

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Bibliographic Details
Published in:Numerical methods for partial differential equations 2006-09, Vol.22 (5), p.1090-1105
Main Authors: Alouges, François, Bonnaillie-Noël, Virginie
Format: Article
Language:English
Subjects:
Mathematics
mesh-refinement technique
natural gauge-invariant method
Numerical Analysis
Schrödinger operator
superconductivity
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Summary:Motivated by questions arising in the theory of superconductivity, we are interested in the study of the fundamental state of the Schrödinger 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. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
ISSN:0749-159X
1098-2426
DOI:10.1002/num.20137