Stabilizer-free weak Galerkin method and its optimal [formula omitted] 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 Pk(K), Pk(e), [Pj(K)]2 to approximate the interior, edge, and discrete weak gradient spaces on each element K and edge e⊂∂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 L2 error estimates. Several numerical experiments are conducted to demonstrate our theoretical findings, where Oseen iteration is utilized for the nonlinear coupling terms.