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Linearized Conservative Finite Element Methods for the Nernst–Planck–Poisson Equations

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Abstract

The aim of this paper is to present and study new linearized conservative schemes with finite element approximations for the Nernst–Planck–Poisson equations. For the linearized backward Euler FEM, an optimal \(L^2\) error estimate is provided almost unconditionally (i.e., when the mesh size h and time step \(\tau \) are less than a small constant). Global mass conservation and electric energy decay of the schemes are also proved. Extension to second-order time discretizations is given. Numerical results in both two- and three-dimensional spaces are provided to confirm our theoretical analysis and show the optimal convergence, unconditional stability, global mass conservation and electric energy decay properties of the proposed schemes.

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Acknowledgements

Huadong Gao was supported in part by a grant from the National Natural Science Foundation of China (NSFC) under Grant No. 11501227. Dongdong He was supported in part by a Grant from the National Natural Science Foundation of China (NSFC) under Grant No. 11402174. The authors would like to thank Dr. Weifeng Qiu for useful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Huadong Gao

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, People’s Republic of China

    Dongdong He

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  1. Huadong Gao
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  2. Dongdong He
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Correspondence to Dongdong He.

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Gao, H., He, D. Linearized Conservative Finite Element Methods for the Nernst–Planck–Poisson Equations. J Sci Comput 72, 1269–1289 (2017). https://doi.org/10.1007/s10915-017-0400-4

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  • DOI: https://doi.org/10.1007/s10915-017-0400-4

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