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Mixed Finite Element Method for Modified Poisson–Nernst–Planck/Navier–Stokes Equations

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Abstract

In this paper, a complete mixed finite element method is developed for a modified Poisson–Nernst–Planck/Navier–Stokes (PNP/NS) coupling system, where the original Poisson equation in PNP system is replaced by a fourth-order elliptic equation to more precisely account for electrostatic correlations in a simplified form of the Landau–Ginzburg-type continuum model. A stabilized mixed weak form is defined for each equation of the modified PNP/NS model in terms of primary variables and their corresponding vector-valued gradient variables, based on which a stable Stokes-pair mixed finite element is thus able to be utilized to discretize all solutions to the entire modified PNP/NS model in the framework of Stokes-type mixed finite element approximation. Semi- and fully discrete mixed finite element schemes are developed and are analyzed for the presented modified PNP/NS equations, and optimal convergence rates in energy norms are obtained for both schemes. Numerical experiments are carried out to validate all attained theoretical results.

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Acknowledgements

M. He is partially supported by Natural Science Foundation of Zhejiang Province, China (Nos. LY21A010011 and LQ19A010009). P. Sun was supported by a Grant from the Simons Foundation (MPS-706640, PS).

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Authors and Affiliations

  1. School of Sciences, Hangzhou Dianzi University, Hangzhou, Zhejiang, China

    Mingyan He

  2. Department of Mathematical Sciences, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV, 89154, USA

    Pengtao Sun

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  1. Mingyan He
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He, M., Sun, P. Mixed Finite Element Method for Modified Poisson–Nernst–Planck/Navier–Stokes Equations. J Sci Comput 87, 80 (2021). https://doi.org/10.1007/s10915-021-01478-z

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