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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
Article

Abstract

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.

Keywords

Discontinuous Galerkin Hybridization Maxwell equations Superconvergence Simplicial mesh General polyhedral mesh 

Mathematics Subject Classification

65N15 65N30 

Notes

Acknowledgments

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.

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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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