Stabilizer-free weak Galerkin method and its optimal L2 error estimates for the time-dependent Poisson—Nernst–Planck problem

Abstract

This paper concerns a backward Euler stabilizer-free weak Galerkin finite element method (SFWG-FEM) for the time-dependent Poisson–Nernst–Planck (TD-PNP) problem. The scheme we propose utilizes spaces $P_k\left(K\right)$, $P_k\left(e\right)$, ${\left[P_j\left(K\right)\right]}^2$ to approximate the interior, edge, and discrete weak gradient spaces on each element $K$ and edge $e\subset \partial K$, respectively. The proposed method is in a simple format similar to the regular finite element method, compatible with polygonal meshes, flexible in approximation function space, and unconditionally stable in time. Based on a rigorous analysis of a weak Galerkin Ritz projection error, which is derived by a dual problem, the superconvergence of the Ritz projection error estimates in energy norm results in optimal $L^2$ error estimates. Several numerical experiments are conducted to demonstrate our theoretical findings, where Oseen iteration is utilized for the nonlinear coupling terms.