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

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 regularity of the linearized system derived by taking G-derivatives of the nonlinear system, we show the robust relationship between the error of solution and the a posteriori error estimator. Numerical experiments are given to validate the efficiency of the a posteriori error estimator and demonstrate the expected rate of convergence. In further tests, adaptive mesh refinements for geometrical singularities and boundary layer effects are successfully observed.