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A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations

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Abstract

We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The computational system is decoupled to smaller systems by using coarse space solutions at each time level, which can speed up the solution process compared with the finite element method combined with the Gummel iteration. We derive the optimal error estimates in L2 norm for both semi- and fully discrete finite element approximations. Based on the a priori error estimates, the error estimates in H1 norm are presented for the two-grid algorithm. The theoretical results indicate this decoupling method can retain the same accuracy as the finite element method. Numerical experiments including the Poisson-Nernst-Planck equations for an ion channel show the efficiency and effectiveness of the decoupling two-grid method.

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Acknowledgments

The authors would like to thank Dr. Chunshen Feng and Dr. Shixin Xu for their valuable discussions on numerical experiments.

Funding

S. Shu was supported by the China NSF (NSFC 11571293). Y. Yang was supported by the China NSF (NSFC 11561016, NSFC 11661027, NSFC 11561015), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation open fund, and the Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University. B. Z. Lu was supported by Science Challenge Program under grant number TZ2016003, National Key Research and Development Program of China (Grant No. 2016YFB0201304), and China NSF (NSFC 21573274, 11771435). R. G. Shen was supported by Postgraduate Scientific Research and Innovation Fund of the Hunan Provincial Education Department (CX2017B268).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Ruigang Shen

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

    Shi Shu

  3. School of Mathematics and Computational Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, 541004, People’s Republic of China

    Ying Yang

  4. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, the National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Benzhuo Lu

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  1. Ruigang Shen
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  2. Shi Shu
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  4. Benzhuo Lu
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Correspondence to Ying Yang.

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Shen, R., Shu, S., Yang, Y. et al. A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations. Numer Algor 83, 1613–1651 (2020). https://doi.org/10.1007/s11075-019-00744-4

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  • DOI: https://doi.org/10.1007/s11075-019-00744-4

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