Adaptive finite element approximation for steady-state Poisson-Nernst-Planck equations

TL;DR

This paper introduces an adaptive finite element method for solving steady-state Poisson-Nernst-Planck equations, focusing on error estimation, solution existence, and boundary layer adaptivity, validated through numerical experiments.

Contribution

It provides a systematic analysis of the nonlinear equations, develops a residual-based a posteriori error estimate, and demonstrates adaptive mesh refinement for complex geometries.

Findings

Efficient a posteriori error estimator validated by numerical experiments

Successful adaptive mesh refinement handling geometrical singularities

Convergence rates consistent with theoretical predictions

Abstract

In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate of the nonlinear system is presented. With the help of Schauder fixed point theorem, we show the solution existence and uniqueness of the linearized system derived by taking $G-$derivatives of the nonlinear system, followed by the proof of the relationship between the error of solution and the a posteriori error estimator $\eta$. Numerical experiments are given to validate the efficiency of the a posteriori error estimator and demonstrate the expected rate of convergence. In the further tests, adaptive mesh refinements for geometrical singularities and boundary layer effects are successfully observed.