Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations

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).