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

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

Received  September 2014 Revised  February 2015 Published  December 2015

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.
Citation: Eduardo Lara, Rodolfo Rodríguez, Pablo Venegas. Spectral approximation of the curl operator in multiply connected domains. Discrete & 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. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. Google Scholar

[2]

E. Beltrami, Considerazioni idrodinamiche,, Il Nuovo Cimento (1877-1894), 25 (1889), 1877. doi: 10.1007/BF02719090. Google Scholar

[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. doi: 10.1137/S0036142901390780. Google Scholar

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011). Google Scholar

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S. Chandrasekhar and P. C. Kendall, On force-free magnetic fields,, Astrophys. J., 126 (1957), 457. doi: 10.1086/146413. Google Scholar

[6]

S. Chandrasekhar and L. Woltjer, On force-free magnetic fields,, Proc. Nat. Acad. Sci. USA, 44 (1958), 285. doi: 10.1073/pnas.44.4.285. Google Scholar

[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. Google Scholar

[8]

V. Girault and P.-A Raviart, Finite Element Approximations of the Navier-Stokes Equations, Theory and Algorithms,, Springer, (1986). doi: 10.1007/978-3-642-61623-5. Google Scholar

[9]

R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators,, Ann. Mat. Pura Appl., 191 (2012), 431. doi: 10.1007/s10231-011-0189-y. Google Scholar

[10]

E. Lara, Espectro del operador rotacional en dominios no simplemente conexos,, Mathematical Engineering thesis, (2013). Google Scholar

[11]

S. Meddahi and V. Selgas, A mixed-FEM and BEM coupling for a three-dimensional eddy current problem,, $M^2AN$, 37 (2003), 291. doi: 10.1051/m2an:2003027. Google Scholar

[12]

B. Mercier, J. Osborn, J. Rappaz and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods,, Math. Comp., 36 (1981), 427. doi: 10.1090/S0025-5718-1981-0606505-9. Google Scholar

[13]

P. Monk, Finite Element Methods for Maxwell's Equations,, Clarendon Press, (2003). doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar

[14]

R. Rodríguez and P. Venegas, Numerical approximation of the spectrum of the curl operator,, Math. Comp., 83 (2014), 553. doi: 10.1090/S0025-5718-2013-02745-7. Google Scholar

[15]

L. Woltjer, A theorem on force-free magnetic fields,, Proc. Natl. Acad. Sci. USA, 44 (1958), 489. doi: 10.1073/pnas.44.6.489. Google Scholar

[16]

_________, The crab nebula,, Bull. Astron. Inst. Neth., 14 (1958), 39. Google Scholar

[17]

J. Xiao and Q. Hu, An iterative method for computing Beltrami fields on bounded domains,, Institute of Computational Mathematics and Scientific/Engineering Computing, (2012), 12. Google Scholar

[18]

Z. Yoshida and Y. Giga, Remarks on spectra of operator rot,, Math. Z., 204 (1990), 235. doi: 10.1007/BF02570870. Google Scholar

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. doi: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B. Google Scholar

[2]

E. Beltrami, Considerazioni idrodinamiche,, Il Nuovo Cimento (1877-1894), 25 (1889), 1877. doi: 10.1007/BF02719090. Google Scholar

[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. doi: 10.1137/S0036142901390780. Google Scholar

[4]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer, (2011). Google Scholar

[5]

S. Chandrasekhar and P. C. Kendall, On force-free magnetic fields,, Astrophys. J., 126 (1957), 457. doi: 10.1086/146413. Google Scholar

[6]

S. Chandrasekhar and L. Woltjer, On force-free magnetic fields,, Proc. Nat. Acad. Sci. USA, 44 (1958), 285. doi: 10.1073/pnas.44.4.285. Google Scholar

[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. Google Scholar

[8]

V. Girault and P.-A Raviart, Finite Element Approximations of the Navier-Stokes Equations, Theory and Algorithms,, Springer, (1986). doi: 10.1007/978-3-642-61623-5. Google Scholar

[9]

R. Hiptmair, P. R. Kotiuga and S. Tordeux, Self-adjoint curl operators,, Ann. Mat. Pura Appl., 191 (2012), 431. doi: 10.1007/s10231-011-0189-y. Google Scholar

[10]

E. Lara, Espectro del operador rotacional en dominios no simplemente conexos,, Mathematical Engineering thesis, (2013). Google Scholar

[11]

S. Meddahi and V. Selgas, A mixed-FEM and BEM coupling for a three-dimensional eddy current problem,, $M^2AN$, 37 (2003), 291. doi: 10.1051/m2an:2003027. Google Scholar

[12]

B. Mercier, J. Osborn, J. Rappaz and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods,, Math. Comp., 36 (1981), 427. doi: 10.1090/S0025-5718-1981-0606505-9. Google Scholar

[13]

P. Monk, Finite Element Methods for Maxwell's Equations,, Clarendon Press, (2003). doi: 10.1093/acprof:oso/9780198508885.001.0001. Google Scholar

[14]

R. Rodríguez and P. Venegas, Numerical approximation of the spectrum of the curl operator,, Math. Comp., 83 (2014), 553. doi: 10.1090/S0025-5718-2013-02745-7. Google Scholar

[15]

L. Woltjer, A theorem on force-free magnetic fields,, Proc. Natl. Acad. Sci. USA, 44 (1958), 489. doi: 10.1073/pnas.44.6.489. Google Scholar

[16]

_________, The crab nebula,, Bull. Astron. Inst. Neth., 14 (1958), 39. Google Scholar

[17]

J. Xiao and Q. Hu, An iterative method for computing Beltrami fields on bounded domains,, Institute of Computational Mathematics and Scientific/Engineering Computing, (2012), 12. Google Scholar

[18]

Z. Yoshida and Y. Giga, Remarks on spectra of operator rot,, Math. Z., 204 (1990), 235. doi: 10.1007/BF02570870. Google Scholar

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