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Residual Type a Posteriori Error Estimates for the Time-Dependent Poisson–Nernst–Planck Equations

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Abstract

This paper investigates the residual type a posteriori error estimators for a fully discrete approximation to the solution of the time-dependent Poisson–Nernst–Planck equations, which are widely used to describe the electrodiffusion of ions in biomolecular solutions. The backward Euler scheme is used for the discretization in time and the continuous, piecewise linear triangular finite elements are applied to the space discretization. The main results consist in building error estimators and deriving computable upper and lower bounds on the error estimators. Some numerical experiments confirm the theoretical predictions and show the reliability and efficiency of the error estimators.

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Data Availability Statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

B. Z. Lu was supported by National Key Research and Development Program of Ministry of Science and Technology (Grant No. 2016YFB0201304), and China NSF (NSFC 11771435, 22073110). Y. Yang was supported by the China NSF (NSFC 12161026), Guangxi Natural Science Foundation (2020GXNSFAA159098, 2017GXNSFFA198012, 2020GXNSFBA238022), 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. G. H. Ji was supported by the China NSF (NSFC 11671052, 11871105). W. W. Zhu was supported by Innovation Project of Guangxi Graduate Education (YCSW2019145).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, 541004, Guangxi, People’s Republic of China

    Wanwan Zhu

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

    Ying Yang

  3. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Guanghua Ji

  4. CEMS, LSEC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Benzhuo Lu

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  4. Benzhuo Lu
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Correspondence to Ying Yang or Guanghua Ji.

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Zhu, W., Yang, Y., Ji, G. et al. Residual Type a Posteriori Error Estimates for the Time-Dependent Poisson–Nernst–Planck Equations. J Sci Comput 90, 27 (2022). https://doi.org/10.1007/s10915-021-01702-w

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