Preconditioners and their analyses for edge element saddle-point systems arising from time-harmonic Maxwell’s equations


We derive and propose a family of new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell’s equations in three dimensions. With the new preconditioners, we show that the preconditioned conjugate gradient method can apply for the saddle-point systems when wave numbers are smaller than a positive critical number, while the iterative methods like the preconditioned MINRES may apply when wave numbers are larger than the critical number. The spectral behaviors of the resulting preconditioned systems for some existing and new preconditioners are analyzed and compared, and several two-dimensional numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.

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The authors would like to thank the anonymous referees for their many insightful and constructive comments and suggestions that have helped us improve the structure and results of the paper essentially.


The research of this project was financially supported by the National Natural Science Foundation of China under grants 11571265 and 11471253. The work of J. Zou was substantially supported by Hong Kong RGC General Research Fund (Project 14304517) and NSFC/Hong Kong RGC Joint Research Scheme 2016/17 (Project N_CUHK437/16).

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Correspondence to Jun Zou.

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Liang, Y., Xiang, H., Zhang, S. et al. Preconditioners and their analyses for edge element saddle-point systems arising from time-harmonic Maxwell’s equations. Numer Algor 86, 281–302 (2021).

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  • Time-harmonic Maxwell’s equations
  • Saddle-point system
  • Preconditioners

Mathematics Subject Classification (2010)

  • 65F10
  • 65N22
  • 65N30