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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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  • [1] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci. 21 (1998), no. 9, 823-864 (English, with English and French summaries). MR 1626990 (99e:35037), https://doi.org/10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B
  • [2] E. Beltrami, Considerazioni idrodinamiche, Rend. Inst. Lombardo Acad. Sci. Let., 22 (1889) 122-131. (English translation: Considerations on hydrodynamics, Int. J. Fusion Energy, 3 (1985) 53-57.)
  • [3] Alfredo Bermúdez, Rodolfo Rodríguez, and Pilar Salgado, Numerical analysis of electric field formulations of the eddy current model, Numer. Math. 102 (2005), no. 2, 181-201. MR 2206462 (2006k:65307), https://doi.org/10.1007/s00211-005-0652-z
  • [4] Daniele Boffi, Finite element approximation of eigenvalue problems, Acta Numer. 19 (2010), 1-120. MR 2652780 (2011e:65256), https://doi.org/10.1017/S0962492910000012
  • [5] T. Z. Boulmezaoud and T. Amari, Approximation of linear force-free fields in bounded 3-D domains, Math. Comput. Modelling 31 (2000), no. 2-3, 109-129. MR 1742463 (2000k:78006), https://doi.org/10.1016/S0895-7177(99)00227-7
  • [6] T. Z. Boulmezaoud and T. Amari, A finite-element method for computing nonlinear force-free fields, Math. Comput. Modelling 34 (2001), no. 7-8, 903-920. MR 1858809 (2002h:85001), https://doi.org/10.1016/S0895-7177(01)00108-X
  • [7] Tahar-Zamène Boulmezaoud, Yvon Maday, and Tahar Amari, On the linear force-free fields in bounded and unbounded three-dimensional domains, M2AN Math. Model. Numer. Anal. 33 (1999), no. 2, 359-393 (English, with English and French summaries). MR 1700040 (2000g:35171), https://doi.org/10.1051/m2an:1999121
  • [8] A. Buffa, M. Costabel, and D. Sheen, On traces for $ {\bf H}({\bf curl},\Omega )$ in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845-867. MR 1944792 (2004i:35045), https://doi.org/10.1016/S0022-247X(02)00455-9
  • [9] Jason Cantarella, Dennis DeTurck, Herman Gluck, and Mikhail Teytel, The spectrum of the curl operator on spherically symmetric domains, Phys. Plasmas 7 (2000), no. 7, 2766-2775. MR 1766493 (2001b:76094), https://doi.org/10.1063/1.874127
  • [10] S. Chandrasekhar and L. Woltjer, On force-free magnetic fields, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 285-289. MR 0098625 (20 #5081)
  • [11] S. Chandrasekhar and P. C. Kendall, On force-free magnetic fields, Astrophys. J. 126 (1957), 457-460. MR 0088988 (19,606d)
  • [12] Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. I. The problem of convergence, RAIRO Anal. Numér. 12 (1978), no. 2, 97-112, (English, with French summary). MR 0483400 (58 #3404a)
  • [13] Jean Descloux, Nabil Nassif, and Jacques Rappaz, On spectral approximation. II. Error estimates for the Galerkin method, RAIRO Anal. Numér. 12 (1978), no. 2, 113-119, (English, with French summary). MR 0483401 (58 #3404b)
  • [14] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383 (88b:65129)
  • [15] A. Lakhtakia, Victor Trkal, Beltrami fields and Trkalian flows, Czech J. Phys., 44 (1994) 89-96.
  • [16] Salim Meddahi and Virginia Selgas, A mixed-FEM and BEM coupling for a three-dimensional eddy current problem, M2AN Math. Model. Numer. Anal. 37 (2003), no. 2, 291-318. MR 1991202 (2004f:78007), https://doi.org/10.1051/m2an:2003027
  • [17] B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), no. 154, 427-453. MR 606505 (82b:65108), https://doi.org/10.2307/2007651
  • [18] Peter Monk, Finite element methods for Maxwell's equations, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2003. MR 2059447 (2005d:65003)
  • [19] Edward C. Morse, Eigenfunctions of the curl in cylindrical geometry, J. Math. Phys. 46 (2005), no. 11, 113511, 13. MR 2186786 (2006j:78022), https://doi.org/10.1063/1.2118447
  • [20] J.-C. Nédélec, Mixed finite elements in $ {\bf R}^{3}$, Numer. Math. 35 (1980), no. 3, 315-341. MR 592160 (81k:65125), https://doi.org/10.1007/BF01396415
  • [21] J. B. Taylor, Relaxation of toroidal and generation of reverse magnetic fields, 33 (1974) 1139-1141.
  • [22] V. Trkal, Poznámka k hydrodynamice vazkých tekutin, Časopis pro Pĕstování Mathematiky a Fysiky, 48 (1919) 302-311. (English translation: A note on the hydrodynamics of viscous fluids, Czech J. Phys., 44 (1994) 97-106.)
  • [23] L. Woltjer, A theorem on force-free magnetic fields, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 489-491. MR 0096542 (20 #3025)
  • [24] L. Woltjer, The crab nebula, Bull. Astron. Inst. Neth., 14 (1958) 39-80.
  • [25] Zensho Yoshida and Yoshikazu Giga, Remarks on spectra of operator rot, Math. Z. 204 (1990), no. 2, 235-245. MR 1055988 (92a:35120), https://doi.org/10.1007/BF02570870

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Additional Information

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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