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Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number

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Abstract

In this paper, using the first-order Nédélec conforming edge element space of the second type, we develop and analyze a continuous interior penalty finite element method (CIP-FEM) for the time-harmonic Maxwell equation in the three-dimensional space. Compared with the standard finite element methods, the novelty of the proposed method is that we penalize the jumps of the tangential component of its vorticity field. It is proved that if the penalty parameter is a complex number with negative imaginary part, then the CIP-FEM is well-posed without any mesh constraint. The error estimates for the CIP-FEM are derived. Numerical experiments are presented to verify our theoretical results.

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References

  1. Adams, R.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Ainsworth, M.: Dispersive properties of high order Nédélec/edge element approximation of the time-harmonic Maxwell equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362, 471–491 (2004)

    Article  MathSciNet  Google Scholar 

  3. Brenner, S., Li, F., Sung, L.: A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell equations. Math. Comp. 70, 573–595 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Chen, L.: IFEM: an integrated finite element methods package in MATLAB, Technical Report, University of California at Irvine (2009)

  5. Cuvelier, C., Segal, A., Steenhoven, A.: Finite element methods and Navier-Stokes equations. D. Reidel Publishing Company, Dordrecht (1986)

    Book  Google Scholar 

  6. Douglas, J. Jr, Dupont, T.: Interior penalty procedures for elliptic and parabolic Galerkin methods, Lecture Notes in Phys., 58. Springer, Berlin (1976)

    Google Scholar 

  7. Feng, X., Wu, H.: An absolutely stable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations with large wave number. SIAM J. Numer. Anal. 52, 2356–2380 (2014)

    Article  MathSciNet  Google Scholar 

  8. Feng, X., Wu, H.: hp-discontinuous Galerkin methods for the Helmholtz equation with large wave number. Math. Comp. 80, 1997–2024 (2011)

    Article  MathSciNet  Google Scholar 

  9. Hiptmair, R., Moiola, A.: Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. Math. Models Methods Appl. Sci. 21, 2263–2287 (2011)

    Article  MathSciNet  Google Scholar 

  10. Hiptmair, R., Moiola, A., Perugia, I.: Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comp. 82, 247–268 (2013)

    Article  MathSciNet  Google Scholar 

  11. Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100, 485–518 (2005)

    Article  MathSciNet  Google Scholar 

  12. Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM. J. Numer. Anal. 42, 434–459 (2004)

    Article  MathSciNet  Google Scholar 

  13. Lu, P., Wang, Y., Xu, X.: Regularity results for the time-harmonic Maxwell equations with impedance boundary condition, submitted, 1804.07856.pdf

  14. Melenk, J.M., Sauter, S.: Wavenumber-explicit hp-FEM analysis for Maxwell’s equations with transparent boundary conditions, arXiv:1803.01619 (2018)

  15. Monk, P.: Finite element methods for Maxwell’s equations. Oxford University Press, New York (2003)

    Book  Google Scholar 

  16. Nicaise, S., Tomezyk, J.: Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary, https://hal.archives-ouvertesfr/hal-02063271/ (2019)

  17. Perugia, I., Schötzau, D., Monk, P.: Stabilized interior penalty methods for the timeharmonic Maxwell equations. Comput. Methods Appl. Mech. Engrg. 191, 4675–4697 (2002)

    Article  MathSciNet  Google Scholar 

  18. Wu, H.: Pre-asymptotic error analysis of CIP-FEM and FEM for Helmholtz equation with high wave number. Part I: Linear version. IMA J. Numer. Anal. 34, 1266–1288 (2014)

    Article  MathSciNet  Google Scholar 

  19. Zhong, L., Shu, S., Wittum, G., Xu, J.: Optimal error estimates for nédélec edge elements for time-harmonic Maxwell’s equations. J. Comp. Math. 27, 563–572 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

We thank the editor and the anonymous referees, who meticulously read through the paper and made many helpful suggestions which led to an improved presentation of this paper.

Funding

We would like to note that the work of Haijun Wu was partially supported by the National Natural Science Foundation of China under grants 11525103 and 91630309 and the work of the Xuejun Xu was supported by National Natural Science Foundation of China (Grant Nos. 11671302 and 11871272).

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Authors and Affiliations

  1. School of Mathematics Sciences, Soochow University, Suzhou, 215006, China

    Peipei Lu

  2. Department of Mathematics, Nanjing University, Jiangsu, 210093, China

    Haijun Wu

  3. School of Mathematical Sciences, Tongji University, Shanghai, 200442, China

    Xuejun Xu

  4. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, China

    Xuejun Xu

Authors
  1. Peipei Lu
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  2. Haijun Wu
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  3. Xuejun Xu
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Correspondence to Peipei Lu.

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Communicated by: Jan Hesthaven

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Lu, P., Wu, H. & Xu, X. Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number. Adv Comput Math 45, 3265–3291 (2019). https://doi.org/10.1007/s10444-019-09737-2

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  • DOI: https://doi.org/10.1007/s10444-019-09737-2

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