Journal of Scientific Computing

, Volume 70, Issue 3, pp 1010–1029 | Cite as

A Superconvergent HDG Method for the Maxwell Equations

  • Huangxin Chen
  • Weifeng Qiu
  • Ke Shi
  • Manuel Solano


We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier p is introduced, and the problem becomes the solution of a mixed curl–curl formulation of the Maxwell’s problem. We use polynomials of degree \(k+1\), k, k to approximate \({{\varvec{u}}},\nabla \times {{\varvec{u}}}\) and p respectively. In contrast, we only use a non-trivial subspace of polynomials of degree \(k+1\) to approximate the numerical tangential trace of the electric field and polynomials of degree \(k+1\) to approximate the numerical trace of the Lagrange multiplier on the faces. On the simplicial meshes, we show that the convergence rates for \(\varvec{u}\) and \(\nabla \times \varvec{u}\) are independent of the Lagrange multiplier p. If we assume the dual operator of the Maxwell equation on the domain has adequate regularity, we show that the convergence rate for \(\varvec{u}\) is \(O(h^{k+2})\). From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. Finally, we show that the superconvergence of the HDG method is also derived on general polyhedral elements. Numerical results are given to verify the theoretical analysis.


Discontinuous Galerkin Hybridization Maxwell equations Superconvergence Simplicial mesh General polyhedral mesh 

Mathematics Subject Classification

65N15 65N30 



The work of Huangxin Chen was supported by the NSF of China (Grant No. 11201394) and the Fundamental Research Funds for the Central Universities (Grant No. 20720150005). The work of Weifeng Qiu was partially supported by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302014). Manuel Solano was partially supported by CONICYT-Chile through the FONDECYT Project No. 1160320 and BASAL Project CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matem’atica (CI\(^2\)MA), Universidad de Concepción, and by CONICYT Project Anillo ACT1118 (ANANUM). As a convention the names of the authors are alphabetically ordered. All authors contributed equally in this article.


  1. 1.
    Adams, R.: Sobolev Spaces. Academic Press, New York (1975)MATHGoogle Scholar
  2. 2.
    Bonito, A., Guermond, J.-L., Luddens, F.: Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains. J. Math. Anal. Appl. 408, 498–512 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bonito, A., Guermond, J.-L., Luddens, F.: An interior penalty method with \(C^0\) finite elements for the approximation of the Maxwell equations in heterogeneous media: convergence analysis with minimal regularity (2014). arXiv:1402.4454
  4. 4.
    Brenner, S., Li, F., Sung, L.: A locally divergence-free interior penalty method for two-dimensional curl–curl problems. SIAM J. Numer. Anal. 42, 1190–1211 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brezzi, F., Douglas, J., Duran, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefMATHGoogle Scholar
  7. 7.
    Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588–610 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47, 1092–1125 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fu, Z., Gatica, L.F., Sayas, F.-J.: Algorithm 949: MATLAB tools for HDG in Three Dimensions. ACM Trans. Math. Softw., 41, 3, Article 20, (2015)Google Scholar
  12. 12.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–239 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hiptmair, R., Moiola, A., Perugia, I.: Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82, 247–268 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Houston, P., Perugia, I., Schötzau, D.: Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal. 42, 434–459 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Houston, P., Perugia, I., Schneebeli, A., Schötzau, D.: Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math. 100, 485–518 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lehrenfeld, C.: Hybrid discontinuous Galerkin methods for solving incompressible flow problems. Diplomingenieur thesis (2010)Google Scholar
  17. 17.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003)CrossRefMATHGoogle Scholar
  18. 18.
    Mu, L., Wang, J., Ye, X., Zhang, S.: A weak Galerkin finite element method for the Maxwell equations. J. Sci. Comput. 65, 363–386 (2015)Google Scholar
  19. 19.
    Nédélec, J.: Mixed finite elements in \(R^3\). Numer. Math. 35, 315–341 (1980)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nédélec, J.: A new family of mixed finite elements in \(R^3\). Numer. Math. 50, 57–81 (1986)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nguyen, N.C., Peraire, J., Cockburn, B.: Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations. J. Comput. Phys. 230, 7151–7175 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Perugia, I., Schötzau, D., Monk, P.: Stabilized interior penalty methods for the time harmonic Maxwell equations. Comput. Methods Appl. Mech. Eng. 191, 4675–4697 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Perugia, I., Schötzau, D.: The hp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations. Math. Comput. 72, 1179–1214 (2003)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Qiu, W., Shen, J., Shi, K.: An HDG method for linear elasticity with strong symmetric stresses. (submitted) arXiv:1312.1407
  25. 25.
    Qiu, W., Shi, K.: An HDG method for convection diffusion equation. J. Sci. Comput. 66, 346–357 (2016)Google Scholar
  26. 26.
    Qiu, W., Shi, K.: A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. (2016). doi: 10.1093/imanum/drv067 MathSciNetGoogle Scholar
  27. 27.
    Zhong, L., Shu, S., Wittum, G., Xu, J.: Optimal error estimates for Nédélec edge elements for time-harmonic Maxwells equations. J. Comput. Math. 27, 563–572 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Huangxin Chen
    • 1
  • Weifeng Qiu
    • 2
  • Ke Shi
    • 3
  • Manuel Solano
    • 4
  1. 1.School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific ComputingXiamen UniversityFujianChina
  2. 2.Department of MathematicsCity University of Hong KongKowloonHong Kong, China
  3. 3.Department of Mathematics and StatisticsOld Dominion UniversityNorfolkUSA
  4. 4.Departamento de Ingeniería Matemática and Centro de Investigación en Ingeniería Matemática (CI²MA)Universidad de ConcepciónConcepciónChile

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