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A discrete weighted Helmholtz decomposition and its application

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Abstract

We propose a discrete weighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a weighted \(L^2\) inner product and stable uniformly with respect to the jumps in the discontinuous weight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxwell system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner.

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Acknowledgments

The authors wish to thank the anonymous referee for many constructive and insightful comments which have led to a great improvement of the results and organization of the paper, and also thank Dr. Junxian Wang and Dr. Chunsheng Feng from Xiangtan University for their help with the numerical experiments.

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Authors and Affiliations

  1. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematical and System Sciences, The Chinese Academy of Sciences, Beijing, 100080, China

    Qiya Hu

  2. Department of Mathematics, Xiangtan University, Hunan, 411105, China

    Shi Shu

  3. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    Jun Zou

Authors
  1. Qiya Hu
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  2. Shi Shu
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  3. Jun Zou
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Corresponding author

Correspondence to Jun Zou.

Additional information

The work of Q. Hu was supported by the Major Research Plan of Natural Science Foundation of China G91130015, the Key Project of Natural Science Foundation of China G11031006 and National Basic Research Program of China G2011309702. The work of S. Shu was supported by NSFC Projects G91130002 and G11171281, the Project of Scientific Research Fund of Hunan Provincial Education Department (12A138) and the Program for Changjiang Scholars and Innovative Research Team in University of China (No. IRT1179). The work of J. Zou was substantially supported by Hong Kong RGC grant (Project 405110) and CUHK Focused Investment Scheme 2012/2014.

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Hu, Q., Shu, S. & Zou, J. A discrete weighted Helmholtz decomposition and its application. Numer. Math. 125, 153–189 (2013). https://doi.org/10.1007/s00211-013-0536-6

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  • DOI: https://doi.org/10.1007/s00211-013-0536-6

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