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An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

Part of: Spectral theory and eigenvalue problems Partial differential equations, boundary value problems Equations of mathematical physics and other areas of application General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  01 February 2016

Ye Li
Ye Li*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
*
*Corresponding author. Email address:yeli11@mails.tsinghua.edu.cn (Y.Li)
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Abstract

In this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

Keywords

Adaptive finite elementhybrid basisnonlinear eigenvalue problemsingularly perturbed problemBose-Einstein condensationuniform convergence

MSC classification

Secondary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 35Q55: NLS-like equations (nonlinear Schrödinger) 65N25: Eigenvalue problems 65N06: Finite difference methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 81Q05: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 2 , February 2016 , pp. 442 - 472
Copyright
Copyright © Global-Science Press 2016 

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