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Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem

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Abstract

Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. An error estimate in H1 norm is obtained for a piecewise finite element approximation to the solution of the nonlinear steady-state Poisson-Nernst-Planck equations. Some superconvergence results are also derived by using the gradient recovery technique for the equations. Numerical results are given to validate the theoretical results. It is also numerically illustrated that the gradient recovery technique can be successfully applied to the computation of the practical ion channel problem to improve the efficiency of the external iteration and save CPU time.

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Acknowledgments

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

Funding

Y. Yang was supported by the National Natural Science Foundation of China (Nos. 11561016, 11701119, 11771105), the Guangxi Natural Science Foundation (2020GXNSFAA159098, 2017GXNSFFA198012), the Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation open project fund, and the Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University. C. Liu was partially supported by NSF grant # 1759535 and the United States - Israel Binational Science Foundation (BSF) # 2024246. B. Z. Lu was supported by the National Key Research and Development Program of China (2016YFB0201304), the Science Challenge Program (No. TZ2016003), and the National Natural Science Foundation of China (No. 11771435). L. Q. Zhong was supported by the National Natural Science Foundation of China (Nos. 11671159, 12071160), the Guangdong Basic and Applied Basic Research Foundation (No. 2019A1515010724), the Characteristic Innovation Projects of Guangdong Colleges and Universities, China (No. 2018KTSCX044), and the General Project topic of Science and Technology in Guangzhou, China (No. 201904010117).

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Authors and Affiliations

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

    Ying Yang

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Ming Tang & Liuqiang Zhong

  3. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, USA

    Chun Liu

  4. LSEC, 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. Ying Yang
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  2. Ming Tang
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  3. Chun Liu
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  4. Benzhuo Lu
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  5. Liuqiang Zhong
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Correspondence to Liuqiang Zhong.

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Communicated by: Long Chen

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Yang, Y., Tang, M., Liu, C. et al. Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem. Adv Comput Math 46, 78 (2020). https://doi.org/10.1007/s10444-020-09819-6

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  • DOI: https://doi.org/10.1007/s10444-020-09819-6

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