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Adaptive finite element approximation for steady-state Poisson-Nernst-Planck equations

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Abstract

In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate of the nonlinear system is presented. With the regularity of the linearized system derived by taking G-derivatives of the nonlinear system, we show the robust relationship between the error of solution and the a posteriori error estimator. Numerical experiments are given to validate the efficiency of the a posteriori error estimator and demonstrate the expected rate of convergence. In further tests, adaptive mesh refinements for geometrical singularities and boundary layer effects are successfully observed.

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Funding

T. Hao and X. Xu received financial support from NSFC (No. 11671302). M. Ma received financial support from NSFC (No. 11701428), “Chen Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation, and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Tingting Hao, Manman Ma & Xuejun Xu

  2. HUA LOOKENG Honors College, Changzhou University, Changzhou, 213164, China

    Tingting Hao

  3. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Xuejun Xu

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  1. Tingting Hao
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Correspondence to Manman Ma.

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Communicated by: Paul Houston

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Hao, T., Ma, M. & Xu, X. Adaptive finite element approximation for steady-state Poisson-Nernst-Planck equations. Adv Comput Math 48, 49 (2022). https://doi.org/10.1007/s10444-022-09938-2

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