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An absolutely stable $ hp$-HDG method for the time-harmonic Maxwell equations with high wave number


Authors: Peipei Lu, Huangxin Chen and Weifeng Qiu
Journal: Math. Comp. 86 (2017), 1553-1577
MSC (2010): Primary 65N12, 65N15, 65N30
DOI: https://doi.org/10.1090/mcom/3150
Published electronically: October 27, 2016
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Abstract: We present and analyze a hybridizable discontinuous Galerkin (HDG) method for the time-harmonic Maxwell equations. The divergence-free condition is enforced on the electric field, then a Lagrange multiplier is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell's problem. The method is shown to be an absolutely stable HDG method for the indefinite time-harmonic Maxwell equations with high wave number. By exploiting the duality argument, the dependence of convergence of the HDG method on the wave number $ \kappa $, the mesh size $ h$ and the polynomial order $ p$ is obtained. Numerical results are given to verify the theoretical analysis.


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Additional Information

Peipei Lu
Affiliation: School of Mathematics Sciences, Soochow University, Suzhou, 215006, People’s Republic of China
Email: pplu@suda.edu.cn

Huangxin Chen
Affiliation: School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Fujian, 361005, People’s Republic of China
Email: chx@xmu.edu.cn

Weifeng Qiu
Affiliation: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China
Email: weifeqiu@cityu.edu.hk

DOI: https://doi.org/10.1090/mcom/3150
Keywords: Hybridizable discontinuous Galerkin method, time-harmonic Maxwell equations, Lagrange multiplier, high wave number
Received by editor(s): April 2, 2015
Received by editor(s) in revised form: October 20, 2015, and January 20, 2016
Published electronically: October 27, 2016
Additional Notes: The work of the first author was supported by the NSF of China (Grant No.11401417), the Program of Natural Science Research of Jiangsu Higher Education Institutions of China (Grant No. 14KJB110021) and Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems (No. 201404).
The work of the second author 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 the third author was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302014).
The third author is the corresponding author. All authors contributed equally in this paper.
Article copyright: © Copyright 2016 American Mathematical Society