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A structure-preserving finite element discretization for the time-dependent Nernst-Planck equation

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Abstract

It is still a challenging task to get a satisfying numerical solution to the time-dependent Nernst-Planck (NP) equation, which satisfies the following three physical properties: solution nonnegativity, total mass conservation, and energy dissipation. In this work, we propose a structure-preserving finite element discretization for the time-dependent NP equation combining a reformulated Jordan-Kinderlehrer-Otto (JKO) scheme and Scharfetter-Gummel (SG) approximation. The JKO scheme transforms a partial differential equation solution problem into an optimization problem. Our finite element discretization strategy with the SG stabilization technique and the Fisher information regularization term in the reformulated JKO scheme can guarantee the convexity of the discrete objective function in the optimization problem. In this paper, we prove that our scheme can preserve discrete solution nonnegativity, maintain total mass conservation, and preserve the decay property of energy. These properties are all validated with our numerical experiments. Moreover, the later numerical results show that our scheme performs better than the traditional Galerkin method with linear Lagrangian basis functions in kee** the above physical properties even when the convection term is dominant and the grid is coarse.

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Acknowledgements

This work was supported by the National Key Research and Development Program of Ministry of Science and Technology under Grant 2016YFB0201304; the China NSF under Grants 11771435 and 22073110; Strategic Priority Research Program of the Chinese Academy of Sciences under Grant XDB36000000; National Natural Science Foundation under Grant 32027801.

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Authors and Affiliations

  1. State Key Laboratory of Scientific and Engineering Computing, National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Bei**g, 100190, People’s Republic of China

    Qianru Zhang & Benzhuo Lu

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Bei**g, 100049, People’s Republic of China

    Qianru Zhang & Benzhuo Lu

  3. Laboratory of Theoretical and Computational Nanoscience, CAS Key Laboratory of Nanophotonic Materials and Devices, CAS Center for Excellence in Nanoscience, Bei**g Key Laboratory of Ambient Particles Health Effects and Prevention Techniques, National Center for Nanoscience and Technology, Chinese Academy of Sciences, Bei**g, 100190, People’s Republic of China

    Bin Tu & Qiaojun Fang

  4. University of Chinese Academy of Sciences, No. 19A Yuquan Road, Bei**g, 100049, People’s Republic of China

    Qiaojun Fang

  5. Sino-Danish Center for Education and Research, Bei**g, 101408, People’s Republic of China

    Qiaojun Fang

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  1. Qianru Zhang
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  2. Bin Tu
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  4. Benzhuo Lu
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Correspondence to Bin Tu or Benzhuo Lu.

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Zhang, Q., Tu, B., Fang, Q. et al. A structure-preserving finite element discretization for the time-dependent Nernst-Planck equation. J. Appl. Math. Comput. 68, 1545–1564 (2022). https://doi.org/10.1007/s12190-021-01571-4

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