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Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow

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Abstract

Based on the partition of unity method (PUM), a local and parallel finite element method is designed and analyzed for solving the stationary incompressible magnetohydrodynamics (MHD). The key idea of the proposed algorithm is to first solve the nonlinear system on a coarse mesh, divide the globally fine grid correction into a series of locally linearized residual problems on some subdomains derived by a class of partition of unity, then compute the local subproblems in parallel, and obtain the globally continuous finite element solution by assembling all local solutions together by the partition of unity functions. The main feature of the new method is that the partition of unity provide a flexible and controllable framework for the domain decomposition. Finally, the efficiency of our theoretical analysis is tested by numerical experiments.

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Acknowledgments

The author would like to thank the referees and the editor for the helpful suggestions. The research was supported by the National Natural Science Foundation of China (Grant No: 11701151,11771348,11401174), and the Major Research and Development Program of China (Grant No. 2016YFB0200901).

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100191, People’s Republic of China

    Xiaojing Dong

  2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Xiaojing Dong

  3. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Yinnian He & Yuhong Zhang

  4. The 20th Research Institute of China Electronics Technology Group Corporation, Xi’an, 710068, People’s Republic of China

    Hongbo Wei

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  1. Xiaojing Dong
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  2. Yinnian He
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Correspondence to Yinnian He.

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Communicated by: Ilaria Perugia

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Dong, X., He, Y., Wei, H. et al. Local and parallel finite element algorithm based on the partition of unity method for the incompressible MHD flow. Adv Comput Math 44, 1295–1319 (2018). https://doi.org/10.1007/s10444-017-9582-4

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  • DOI: https://doi.org/10.1007/s10444-017-9582-4

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