Spectral approximation of the curl operator in multiply connected domains

Previous Article
A note on the Navier-Stokes IBVP with small data in $L^n$
DCDS-S Home
This Issue
Next Article
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
Related Papers

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.

[1]	P. Robert Kotiuga. On the topological characterization of near force-free magnetic fields, and the work of late-onset visually-impaired topologists. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 215-234. doi: 10.3934/dcdss.2016.9.215
[2]	Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080
[3]	Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181
[4]	Mohamed Badreddine, Thomas K. DeLillo, Saman Sahraei. A Comparison of some numerical conformal mapping methods for simply and multiply connected domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 55-82. doi: 10.3934/dcdsb.2018100
[5]	Jürgen Scheurle, Stephan Schmitz. A criterion for asymptotic straightness of force fields. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 777-792. doi: 10.3934/dcdsb.2010.14.777
[6]	Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial and Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[7]	Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
[8]	Uwe Helmke, Jens Jordan, Julia Lieb. Probability estimates for reachability of linear systems defined over finite fields. Advances in Mathematics of Communications, 2016, 10 (1) : 63-78. doi: 10.3934/amc.2016.10.63
[9]	Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143-149. doi: 10.3934/amc.2018009
[10]	Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281
[11]	Alexander Gorodnik. Open problems in dynamics and related fields. Journal of Modern Dynamics, 2007, 1 (1) : 1-35. doi: 10.3934/jmd.2007.1.1
[12]	Jiayu Han. Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3195-3212. doi: 10.3934/dcdsb.2018281
[13]	Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic and Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669
[14]	Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031
[15]	Francesco Cellarosi, Ilya Vinogradov. Ergodic properties of $k$-free integers in number fields. Journal of Modern Dynamics, 2013, 7 (3) : 461-488. doi: 10.3934/jmd.2013.7.461
[16]	Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107
[17]	Igor E. Shparlinski. On some dynamical systems in finite fields and residue rings. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 901-917. doi: 10.3934/dcds.2007.17.901
[18]	Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[19]	Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203
[20]	Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Eduardo Lara Rodolfo Rodríguez Pablo Venegas
on Google Scholar
Eduardo Lara Rodolfo Rodríguez Pablo Venegas

[Back to Top]