Skip to main content
SpringerLink
Log in

A plane wave least squares method for the Maxwell equations in anisotropic media

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, we first consider the time-harmonic Maxwell equations with Dirichlet boundary conditions in three-dimensional anisotropic media, where the coefficients of the equations are general symmetric positive definite matrices. By using scaling transformations and coordinate transformations, we build the desired stability estimates between the original electric field and the transformed nonphysical field on the condition number of the anisotropic coefficient matrix. More importantly, we prove that the resulting approximate solutions generated by plane wave least squares (PWLS) methods have the nearly optimal L2 error estimates with respect to the condition number of the coefficient matrix. Finally, numerical results verify the validity of the theoretical results, and the comparisons between the proposed PWLS method and the existing PWDG method are also provided.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Antunes, P.: A numerical algorithm to reduce ill-conditioning in meshless methods for the Helmholtz equation. Numer. Algor. 79, 879–897 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cessenat, O.: Application D’une Nouvelle Formulation Variationnelle Aux équations D’ondes Harmoniques Problèmes De Helmholtz 2D et de Maxwell 3D, Ph.D. Thesis, Université Paris IX Dauphine (1996)

  3. Cessenat, O., Despres, B.: Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35, 255–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Congreve, S., Houston, P., Perugia, I.: Adaptive refinement for hp −version Trefftz discontinuous Galerkin methods for the homogeneous Helmholtz problem. Adv. Comput. Math. https://doi.org/10.1007/s10444-018-9621-9 (2018)

  5. Gabard, G.: Discontinuous Galerkin methods with plane waves for time-harmonic problems. J. Comput. Phys. 225, 1961–1984 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Geuzaine, C., Remacle, J.-F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Meth. Engng 79, 1309–1331 (2009)

    Article  MATH  Google Scholar 

  7. Gittelson, C., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the h-version, ESAIM: m2AN. Math. Model. Numer. Anal. 43, 297–331 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hiptmair, R., Moiola, A., Perugia, I.: Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49, 264–284 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hiptmair, R., Moiola, A., Perugia, I.: Stability results for the time-harmonic Maxwell equationis with impedance boundary conditions. Math. Mod. Meth. Appl. Sci. 21, 2263–2287 (2011)

    Article  MATH  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  MATH  Google Scholar 

  11. Hiptmair, R., Moiola, A., Perugia, I.: A survey of Trefftz methods for the Helmholtz equation. In: Building bridges: connections and challenges in modern approaches to numerical partial differential equations, pp 237–279. Springer International Publishing, New York (2015)

  12. Hu, Q., Li, X.: Efficient multilevel preconditioners for three-dimensional plane wave Helmholtz systems with large wave numbers. Multiscale Model Simul. 15(3), 1242–1266 (2017)

    Article  MathSciNet  Google Scholar 

  13. Hu, Q., Song, R.: A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations, ESAIM: m2AN. Math. Model. Numer. Anal. 53, 85–103 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu, Q., Yuan, L.: A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations. Int. J. Numer. Anal. Model. 11, 587–607 (2014)

    MathSciNet  Google Scholar 

  15. Hu, Q., Yuan, L.: A plane wave Least-Squares method for Time-Harmonic maxwell’s equations in absorbing media. SIAM J. Sci. Comput. 36, A1911–A1936 (2014)

    Article  MathSciNet  Google Scholar 

  16. Hu, Q., Yuan, L.: A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. Adv. Comput. Math. 44, 245–275 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu, Q., Zhang, H.: Substructuring preconditioners for the systems arising from plane wave discretization of Helmholtz equations. SIAM J. Sci. Comput. 38, A2232–A2261 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huttunen, T., Malinen, M., Monk, P.: Solving Maxwell’s equations using the ultra weak variational formulation. J. Comput. Phys. 223, 731–758 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Huttunen, T., Monk, P.: The use of plane waves to approximate wave propagation in anisotropic media. J. Comput. Math. 25, 350–367 (2007)

    MathSciNet  Google Scholar 

  20. Kapita, S., Monk, P., Warburton, T.: Residual-based adapitivity and PWDG methods for the Helmholtz equation. SIAM J. Sci. Comput. 37, A1525–1553 (2015)

    Article  MATH  Google Scholar 

  21. Luostari, T.: Non-polynomial approximation methods in acoustics and elasticity, Ph.D. Thesis, University of Eastern Finland. Available at https://core.ac.uk/download/pdf/19163531.pdf (2013)

  22. Melenk, J.: On approximation in meshless methods, frontiers of numerical analysis, universitext, pp 65–141. Springer, New York (2005)

    Google Scholar 

  23. Moiola, A., Hiptmair, R., Perugia, I.: Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62, 809–837 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon Press, Oxford (2003)

    Book  MATH  Google Scholar 

  25. Monk, P., Wang, D.: A least-squares method for the Helmholtz equation. Comput. Meth. Appl. Mech. Engng. 175, 121–136 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peng, J., Wang, J., Shu, S.: Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations. Int. J. Numer. Methods Eng. 116, 683–707 (2018)

    Article  MathSciNet  Google Scholar 

  27. Peng, J., Shu, S., Wang, J., Zhong, L.: Adaptive-Multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems. J. Comput. Appl. Math. arXiv:1909.04426 (2019)

  28. Sloane, N.: Tables of spherical codes (with collaboration of R.H. Hardin, W.D. Smith and others) published electronically at http://www2.research.att.com/njas/packings (2000)

  29. Trefftz, E.: Ein gegenstück zum Ritzschen Verfahren. Sec. Inte. Cong. Appl. Mech. 131–137 (1926)

  30. Yuan, L., Hu, Q.: Error analysis of the plane wave discontinuous Galerkin method for Maxwell’s equations in anisotropic media. Commun. Comput. Phys. 25, 1496–1522 (2019)

    Article  MathSciNet  Google Scholar 

  31. Yuan, L, Hu, Q.: Plane wave discontinuous Galerkin methods for the Helmholtz equation and Maxwell equations in anisotropic media, available at arXiv:1909.05997v2

  32. Yuan, L., Hu, Q., Hengbin, A: Parallel preconditioners for plane wave Helmholtz and Maxwell systems with large wave numbers. Int. J. Numer. Anal. Model 13, 802–819 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank the anonymous reviewers who give many insightful comments to improve the presentation of this paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Long Yuan

Authors
  1. Long Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Long Yuan.

Additional information

Publisher’s note

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

This author was supported by China NSF under the grant 11501529, Qinddao applied basic research project under grant 17-1-1-9-jch.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, L. A plane wave least squares method for the Maxwell equations in anisotropic media. Numer Algor 87, 873–894 (2021). https://doi.org/10.1007/s11075-020-00991-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-020-00991-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation