Skip to main content
SpringerLink
Log in

Numerical analysis of a finite element method for the electromagnetic concentrator model

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the electromagnetic concentrator model obtained through transformation optics. This model is formed by a system of coupled time-dependent Maxwell’s equations with three unknowns, which makes the analysis and simulation much more challenging compared to the standard Maxwell equations. In our previous work (W. Yang, J. Li, Y. Huang, and B. He, Commun. Comput. Phys., 25(1), pp. 135–154, 2019), we proposed a finite element time-domain (FETD) method with edge elements for solving this model efficiently without any theoretical analysis. Here, we provide a rigorous analysis for the mathematical modelling equations and the FETD method proposed there.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banks, H., Bokil, V., Gibson, N.: Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media. Numer. Methods Partial Differ Equ. 25, 885–917 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bao, G., Li, P., Wu, H.: An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures. Math. Comput. 79, 1–34 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. Math. Model. Numer. Anal. 34, 159–182 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boffi, D., Fernandez, P., Perugia, I.: Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM. J. Numer. Anal. 36, 1264–1290 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brenner, S.C., Gedicke, J., Sung, L.-Y.: Hodge decomposition for two-dimensional time-harmonic Maxwell’s equations: impedance boundary condition. Math. Methods Appl. Sci. 40(2), 370–390 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, W.: Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  7. Cao, L., Zhang, Y., Allegretto, W., Lin, Y.: Multiscale asymptotic method for Maxwell’s equations in composite materials. SIAM J. Numer. Anal. 47, 4257–4289 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carstensen, C., Demkowicz, L., Gopalakrishnan, J.: Breaking spaces and forms for the DPG method and applications including Maxwell equations. Comput. Math. Appl. 72(3), 494–522 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37, 1542–1570 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chung, E.T., Engquist, B.: Convergence analysis of fully discrete finite volume methods for Maxwell’s equations in nonhomogeneous media. SIAM J. Numer. Anal. 43(1), 303–317 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciarlet Jr, P., Zou, J.: Fully discrete finite element approaches for time-dependent Maxwell’s equations. Numer. Math. 82, 193–219 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Demkowicz, L., Kurtz, J., Pardo, D., Paszynski, M., Rachowicz, W., Zdunek, A.: Computing with Hp finite elements. II. Frontiers: Three-dimensional elliptic and maxwell problems with applications. Chapman Hall/CRC, Boca Raton (2007)

    Book  MATH  Google Scholar 

  13. Engheta, N., Ziolkowski, R.W. (eds.): Electromagnetic metamaterials: Physics and Engineering Explorations. Wiley-IEEE Press, New York (2006)

    Google Scholar 

  14. Greenleaf, A., Kurylev, Y., Lassas, M., Uhlmann, G.: Cloaking devices, electromagnetic wormholes, and transformation optics. SIAM Rev. 51, 3–33 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hao, Y., Mittra, R.: FDTD Modeling of metamaterials: theory and applications artech house publishers (2008)

  16. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huang, C., Wang, L.-L.: An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media. Adv. Comput. Math. 45, 707–734 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kohn, R.V., Shen, H., Vogelius, M.S., Weinstein, M.I.: Cloaking via change of variables in electric impedance tomography. Inverse Problems 24(1), 015016 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, J., Huang, Y.: Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer Ser. Comput Math, vol. 43. Springer, New York (2013)

    Google Scholar 

  20. Li, J., Huang, Y., Yang, W., Wood, A.: Mathematical analysis and time domain finite element simulation of carpet cloak. SIAM J. Appl. Math. 74(4), 1136–1151 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, J., Meng, C., Huang, Y.: Improved analysis and simulation of a time-domain carpet cloak model. Comput. Methods Appl Math. 19, 359–378 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Li, J., Shi, C., Shu, C.-W.: Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials. Comput. Math. Appl. 73, 1760–1780 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated Solution of Differential Equations by the Finite Element method: The FEniCS Book. Springer, Berlin (2012)

    MATH  Google Scholar 

  24. Markos, P., Soukoulis, C.M.: Wave propagation: from electrons to photonic crystals and Left-Handed materials. Princeton University Press, Princeton (2008)

  25. Monk, P.: Analysis of a finite element method for Maxwell’s equations. SIAM J. Numer. Anal. 29(3), 714–729 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Monk, P.: Finite element methods for maxwell’s equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  27. Nédélec, J.-C.: Mixed finite elements in \(\mathcal {R}^{3}\). Numer. Math. 35, 315–341 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rahm, M., Schurig, D., Roberts, D.A., Cummer, S.A., Smith, D.R., Pendry, J.B.: Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations. Photonics and Nanostructures-fundamentals and Applications 6(1), 87–95 (2008)

    Google Scholar 

  30. Scheid, C., Lanteri, S.: Convergence of a Discontinuous Galerkin scheme for the mixed time domain Maxwell’s equations in dispersive media. IMA J. Numer. Anal. 33(2), 432–459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, W., Lin, L., Ma, J., Wang, C., Cui, J., Du, C., Luo, X.: Electromagnetic concentrators with reduced material parameters based on coordinate transformation. Opt. Express 16, 11431–11437 (2008)

    Article  Google Scholar 

  32. Xie, Z., Wang, J., Wang, B., Chen, C.: Solving Maxwell’s equation in meta-materials by a CG-DG method. Commun. Comput. Phys. 19(5), 1242–1264 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, W., Li, J., Huang, Y.: Modeling and analysis of the optical black hole in metamaterials by the finite element time-domain method. Comput. Methods Appl. Mech. Engrg. 304, 501–520 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Yang, W., Li, J., Huang, Y.: Mathematical analysis and finite element time domain simulation of arbitrary star-shaped electromagnetic cloaks. SIAM J. Numer. Anal. 56(1), 136–159 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, W., Li, J., Huang, Y., He, B.: Developing finite element methods for simulating transformation optics devices with metamaterials. Commun. Comput. Phys. 25(1), 135–154 (2019)

    Article  MathSciNet  Google Scholar 

  36. Yang, Z., Wang, L.-L., Rong, Z., Wang, B., Zhang, B.: Seamless integration of global Dirichlet-to-Neumann boundary condition and spectral elements for transformation electromagnetics. Comput. Methods Appl. Mech. Engrg. 301, 137–163 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comp. 81(278), 623–642 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We are very grateful to two anonymous referees for their insightful comments which improved our paper.

Funding

This work was partially supported by NSFC projects 11971410, and National Science Foundation under Grant No. DMS-20-11943.

Author information

Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, China

    Yunqing Huang

  2. Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154-4020, USA

    Jichun Li

Authors
  1. Yunqing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jichun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jichun Li.

Additional information

Communicated by: Jan Hesthaven

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Y., Li, J. Numerical analysis of a finite element method for the electromagnetic concentrator model. Adv Comput Math 46, 77 (2020). https://doi.org/10.1007/s10444-020-09817-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10444-020-09817-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation