In this paper, a linearized structure-preserving Galerkin finite element method is investigated for Poisson-Nernst-Planck (PNP) equations. By making full use of the high accuracy estimation of the bilinear element, the mean value technique and rigorously dealing with the coupled nonlinear term, not only the unconditionally optimal error estimate in \(L^2\)-norm but also the unconditionally superclose error estimate in \(H^1\)-norm for the related variables are obtained. Then, the unconditionally global superconvergence error estimate in \(H^1\)-norm is derived by a simple and efficient interpolation post-processing approach, without any coupling restriction condition between the time step size and the space mesh width. Finally, numerical results are provided to confirm the theoretical findings. The numerical scheme preserves the global mass conservation and the electric energy decay, and this work has a great improvement of the error estimate results given in Prohl and Schmuck (Numer. Math. 111, 591–630 2009) and Gao and He (J. Sci. Comput. 72, 1269–1289 2017).

在本文中，研究了泊松-能斯特-普朗克（PNP）方程的线性化保结构伽辽金有限元方法。充分利用双线性元的高精度估计、均值技术以及严格处理耦合非线性项，不仅可以得到\(L^2\) -范数下的无条件最优误差估计，而且可以得到无条件超接近误差获得相关变量的\(H^1\)范数估计。然后，通过简单高效的插值后处理方法导出\(H^1\)范数下的无条件全局超收敛误差估计，且时间步长和空间网格宽度之间没有任何耦合限制条件。最后，提供数值结果来证实理论结果。该数值格式保留了全局质量守恒和电能衰减，并且这项工作对Prohl和Schmuck (Numer. Math. 111 , 591–630 2009)以及Gao和He (J.科学。72，1269-1289 2017）。