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Spectral approximation of the curl operator in multiply connected domains

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

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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), 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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