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On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists
Spectral approximation of the curl operator in multiply connected domains
1. | CI2MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile, Chile |
2. | Department of Mathematics, University of Maryland, College Park, MD 20742, United States |
References:
[1] |
C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.
doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. |
[2] |
E. Beltrami, Considerazioni idrodinamiche, Il Nuovo Cimento (1877-1894), 25 (1889), 212-222; English translation: Considerations on hydrodynamics, Int. J. Fusion Energy, 3 (1985), 53-57.
doi: 10.1007/BF02719090. |
[3] |
A. Bermúdez, R. Rodríguez and P. Salgado, A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations, SIAM J. Numer. Anal., 40 (2002), 1823-1849.
doi: 10.1137/S0036142901390780. |
[4] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[5] |
S. Chandrasekhar and P. C. Kendall, On force-free magnetic fields, Astrophys. J., 126 (1957), 457-460.
doi: 10.1086/146413. |
[6] |
S. Chandrasekhar and L. Woltjer, On force-free magnetic fields, Proc. Nat. Acad. Sci. USA, 44 (1958), 285-289.
doi: 10.1073/pnas.44.4.285. |
[7] |
C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phńomènes successifs de bifurcation, Ann. Sc. Norm. Sup. Pisa, 5 (1978), 28-63. |
[8] |
V. Girault and P.-A Raviart, Finite Element Approximations of the Navier-Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[9] |
R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Ann. Mat. Pura Appl., 191 (2012), 431-457.
doi: 10.1007/s10231-011-0189-y. |
[10] |
E. Lara, Espectro del operador rotacional en dominios no simplemente conexos, Mathematical Engineering thesis, Universidad de Concepción, Chile, 2013. Available from: http://www.ing-mat.udec.cl/~rodolfo/Tesis/Memoria_Titulo_Eduardo_Lara.pdf. |
[11] |
S. Meddahi and V. Selgas, A mixed-FEM and BEM coupling for a three-dimensional eddy current problem, $M^2AN$, Math. Model. Numer. Anal., 37 (2003), 291-318.
doi: 10.1051/m2an:2003027. |
[12] |
B. Mercier, J. Osborn, J. Rappaz and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp., 36 (1981), 427-453.
doi: 10.1090/S0025-5718-1981-0606505-9. |
[13] |
P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003.
doi: 10.1093/acprof:oso/9780198508885.001.0001. |
[14] |
R. Rodríguez and P. Venegas, Numerical approximation of the spectrum of the curl operator, Math. Comp., 83 (2014), 553-577.
doi: 10.1090/S0025-5718-2013-02745-7. |
[15] |
L. Woltjer, A theorem on force-free magnetic fields, Proc. Natl. Acad. Sci. USA, 44 (1958), 489-491.
doi: 10.1073/pnas.44.6.489. |
[16] |
_________, The crab nebula, Bull. Astron. Inst. Neth., 14 (1958), 39-80. |
[17] |
J. Xiao and Q. Hu, An iterative method for computing Beltrami fields on bounded domains, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Report No. ICMSEC-12-15 2012. Available from: http://www.cc.ac.cn/2012reserchreport/201215.pdf. |
[18] |
Z. Yoshida and Y. Giga, Remarks on spectra of operator rot, Math. Z., 204 (1990), 235-245.
doi: 10.1007/BF02570870. |
show all references
References:
[1] |
C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional non-smooth domains, Math. Methods Appl. Sci., 21 (1998), 823-864.
doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. |
[2] |
E. Beltrami, Considerazioni idrodinamiche, Il Nuovo Cimento (1877-1894), 25 (1889), 212-222; English translation: Considerations on hydrodynamics, Int. J. Fusion Energy, 3 (1985), 53-57.
doi: 10.1007/BF02719090. |
[3] |
A. Bermúdez, R. Rodríguez and P. Salgado, A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations, SIAM J. Numer. Anal., 40 (2002), 1823-1849.
doi: 10.1137/S0036142901390780. |
[4] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[5] |
S. Chandrasekhar and P. C. Kendall, On force-free magnetic fields, Astrophys. J., 126 (1957), 457-460.
doi: 10.1086/146413. |
[6] |
S. Chandrasekhar and L. Woltjer, On force-free magnetic fields, Proc. Nat. Acad. Sci. USA, 44 (1958), 285-289.
doi: 10.1073/pnas.44.4.285. |
[7] |
C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phńomènes successifs de bifurcation, Ann. Sc. Norm. Sup. Pisa, 5 (1978), 28-63. |
[8] |
V. Girault and P.-A Raviart, Finite Element Approximations of the Navier-Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[9] |
R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators, Ann. Mat. Pura Appl., 191 (2012), 431-457.
doi: 10.1007/s10231-011-0189-y. |
[10] |
E. Lara, Espectro del operador rotacional en dominios no simplemente conexos, Mathematical Engineering thesis, Universidad de Concepción, Chile, 2013. Available from: http://www.ing-mat.udec.cl/~rodolfo/Tesis/Memoria_Titulo_Eduardo_Lara.pdf. |
[11] |
S. Meddahi and V. Selgas, A mixed-FEM and BEM coupling for a three-dimensional eddy current problem, $M^2AN$, Math. Model. Numer. Anal., 37 (2003), 291-318.
doi: 10.1051/m2an:2003027. |
[12] |
B. Mercier, J. Osborn, J. Rappaz and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp., 36 (1981), 427-453.
doi: 10.1090/S0025-5718-1981-0606505-9. |
[13] |
P. Monk, Finite Element Methods for Maxwell's Equations, Clarendon Press, Oxford, 2003.
doi: 10.1093/acprof:oso/9780198508885.001.0001. |
[14] |
R. Rodríguez and P. Venegas, Numerical approximation of the spectrum of the curl operator, Math. Comp., 83 (2014), 553-577.
doi: 10.1090/S0025-5718-2013-02745-7. |
[15] |
L. Woltjer, A theorem on force-free magnetic fields, Proc. Natl. Acad. Sci. USA, 44 (1958), 489-491.
doi: 10.1073/pnas.44.6.489. |
[16] |
_________, The crab nebula, Bull. Astron. Inst. Neth., 14 (1958), 39-80. |
[17] |
J. Xiao and Q. Hu, An iterative method for computing Beltrami fields on bounded domains, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Report No. ICMSEC-12-15 2012. Available from: http://www.cc.ac.cn/2012reserchreport/201215.pdf. |
[18] |
Z. Yoshida and Y. Giga, Remarks on spectra of operator rot, Math. Z., 204 (1990), 235-245.
doi: 10.1007/BF02570870. |
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