Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

Abstract

This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H1‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L2‐norm in the previous literature. Furthermore, the global superconvergence results in H1‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis.

Nonconforming finite element method for coupled Poisson–Nernst–Planck equations

“…We will take low‐order nonconforming element [7, 12, 16, 18–22] as an example to develop and analyze the semidiscrete and backward Euler fully discrete schemes, and then to derive the superclose and superconvergent error estimates for the variables p i and ψ by use of the special characters of this element mentioned above. We should point out that the proofs in our article are simplified with different mathematics induction assumption, and the conclusions are improved compared with [2, 4, 5, 11, 17, 24]. On the other hand, the superconvergent behavior obtained herein are also valid to other popular nonconforming elements, such as the rotated Q 1 element on square meshes [10] and the constrained rotated Q 1 element on rectangular meshes [6, 9, 13, 15].…”

“…In this section, we present a numerical example as [17, 24] to confirm our theoretical analysis. Let Ω be the unit square [0, 1] 2 , and the final time T = 1.…”

“…There have been much efforts devoted to the theoretical and numerical analysis of PNP equations with different boundary conditions. We refer to [3,8] for the analytical solution, [1] the finite difference method, [25] the finite volume method, and [2,4,5,11,17,24] the FEMs. But only the suboptimal and optimal error estimates of conforming FEs were considered in [2,4,5,11,24], respectively.…”

“…But only the suboptimal and optimal error estimates of conforming FEs were considered in [2,4,5,11,24], respectively. And the superconvergence behavior of (1) was studied for bilinear element in [17] with stronger regularities of the solutions. Later, the same superconvergent error estimates of this system were discussed with linear element in [14] under weaker regularity requirements of p i and 𝜓 compared with [17] .…”

Local Averaging Type a Posteriori Error Estimates for the Nonlinear Steady-state Poisson-Nernst-Planck Equations

“…Gao and He [20] constructed a linearized conservative finite element method to discrete the PNP system with zero Neumann boundary conditions and established unconditionally optimal error estimates in L 2 norm. The superconvergence analysis of finite element method for the time-dependent PNP equations is studied by Shi and Yang in [23]. Besides, in order to obtain the optimal error estimates in L 2 norm for both the electrostatic potential and the ionic concentrations, a mixed finite element method is also studied for PNP equations, see [24,25] for more details.…”