Spectral approximation of the curl operator in multiply connected domains

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On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists

February 2016, 9(1): 235-253. doi: 10.3934/dcdss.2016.9.235

Spectral approximation of the curl operator in multiply connected domains

Eduardo Lara ^1, , Rodolfo Rodríguez ^1, and Pablo Venegas ^2,

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

Received September 2014 Revised February 2015 Published December 2015

Abstract
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A numerical scheme based on Nédélec finite elements has been recently introduced to solve the eigenvalue problem for the curl operator in simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane. However, additional constraints can be added to the eigenvalue problem in order to recover a well-posed problem with a discrete spectrum. Vanishing circulations on each non-bounding cycle in the complement of the domain have been chosen as additional constraints in this paper. A mixed weak formulation including a Lagrange multiplier (that turns out to vanish) is introduced and shown to be well-posed. This formulation is discretized by Nédélec elements, while standard finite elements are used for the Lagrange multiplier. Spectral convergence is proved as well as a priori error estimates. It is also shown how to implement this finite element discretization taking care of these additional constraints. Finally, a numerical test to assess the performance of the proposed methods is reported.

Keywords: linear force-free fields., Beltrami fields, spectrum of the curl operator, Eigenvalue problems, finite elements, multiply connected domains.

Mathematics Subject Classification: Primary: 65N25, 65N30; Secondary: 76M10, 78M1.

Citation: Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 235-253. doi: 10.3934/dcdss.2016.9.235

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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