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Numerical approximation of the spectrum of the curl operator


Authors: Rodolfo Rodríguez and Pablo Venegas
Journal: Math. Comp. 83 (2014), 553-577
MSC (2010): Primary 65N25, 65N30; Secondary 76M10, 78M10
DOI: https://doi.org/10.1090/S0025-5718-2013-02745-7
Published electronically: July 25, 2013
MathSciNet review: 3143684
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Abstract: The aim of this paper is to study the numerical approximation of the eigenvalue problem for the curl operator. The three-dimensional divergence-free eigensolutions of this problem are examples of the so-called Beltrami fields or linear force-free fields, which arise in various physics areas such as solar physics, plasma physics, and fluid mechanics. The present analysis is restricted to bounded simply-connected domains. Finite element discretizations of two weak formulations of the spectral problem are proposed and analyzed. Optimal-order spectral convergence is proved, as well as absence of spurious modes. The results of some numerical tests are also reported.


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Rodolfo Rodríguez
Affiliation: CI$^{2}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: rodolfo@ing-mat.udec.cl

Pablo Venegas
Affiliation: CI$^{2}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Email: pvenegas@ing-mat.udec.cl

DOI: https://doi.org/10.1090/S0025-5718-2013-02745-7
Keywords: Eigenvalue problems, finite elements, spectrum of the curl operator, Beltrami fields, linear force-free fields
Received by editor(s): July 21, 2011
Received by editor(s) in revised form: May 25, 2012
Published electronically: July 25, 2013
Additional Notes: The first author was partially supported by BASAL project CMM, Universidad de Chile (Chile).
The second author was supported by a CONICYT fellowship (Chile).
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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