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 - <p style="text-align: justify;">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.</p>