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Volume 43, Issue 3
Error Analysis of Virtual Element Methods for the Time-Dependent Poisson-Nernst-Planck Equations

Ying Yang, Ya Liu, Yang Liu & Shi Shu

DOI: 10.4208/jcm.2401-m2023-0130

J. Comp. Math., 43 (2025), pp. 731-770.

Published online: 2024-11

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  • Abstract

We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck (PNP) equations, which are a nonlinear coupled system widely used in semiconductors and ion channels. After presenting the semi-discrete scheme, the optimal $H^1$ norm error estimates are presented for the time-dependent PNP equations, which are based on some error estimates of a virtual element energy projection. The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation. The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.

  • Keywords

Virtual element method, Error estimate, Poisson-Nernst-Planck equations, Polygonal meshes, Energy projection, Gummel iteration.

  • AMS Subject Headings

65N15, 65N30, 35K61

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-731, author = {Yang , YingLiu , YaLiu , Yang and Shu , Shi}, title = {Error Analysis of Virtual Element Methods for the Time-Dependent Poisson-Nernst-Planck Equations}, journal = {Journal of Computational Mathematics}, year = {2024}, volume = {43}, number = {3}, pages = {731--770}, abstract = {

We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck (PNP) equations, which are a nonlinear coupled system widely used in semiconductors and ion channels. After presenting the semi-discrete scheme, the optimal $H^1$ norm error estimates are presented for the time-dependent PNP equations, which are based on some error estimates of a virtual element energy projection. The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation. The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2401-m2023-0130}, url = {http://global-sci.org/intro/article_detail/jcm/23557.html} }
TY - JOUR T1 - Error Analysis of Virtual Element Methods for the Time-Dependent Poisson-Nernst-Planck Equations AU - Yang , Ying AU - Liu , Ya AU - Liu , Yang AU - Shu , Shi JO - Journal of Computational Mathematics VL - 3 SP - 731 EP - 770 PY - 2024 DA - 2024/11 SN - 43 DO - http://doi.org/10.4208/jcm.2401-m2023-0130 UR - https://global-sci.org/intro/article_detail/jcm/23557.html KW - Virtual element method, Error estimate, Poisson-Nernst-Planck equations, Polygonal meshes, Energy projection, Gummel iteration. AB -

We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck (PNP) equations, which are a nonlinear coupled system widely used in semiconductors and ion channels. After presenting the semi-discrete scheme, the optimal $H^1$ norm error estimates are presented for the time-dependent PNP equations, which are based on some error estimates of a virtual element energy projection. The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation. The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.

Yang , YingLiu , YaLiu , Yang and Shu , Shi. (2024). Error Analysis of Virtual Element Methods for the Time-Dependent Poisson-Nernst-Planck Equations. Journal of Computational Mathematics. 43 (3). 731-770. doi:10.4208/jcm.2401-m2023-0130
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