Newton Solvers for Drift-Diffusion and Electrokinetic Equations

Abstract

A Newton solver for equations modeling drift-diffusion and electrokinetic phenomena is investigated. For drift-diffusion problems, modeled by the nonlinear Poisson--Nernst--Planck (PNP) equations, the linearization of the model equations is shown to be well-posed. Furthermore, a fast solver for the linearized PNP and electrokinetic equations is proposed and numerically demonstrated to be effective on some physically motivated benchmarks. This work builds on a formulation of the PNP and electrokinetic equations that is investigated in [M. S. Metti, J. Xu, and C. Liu, J. Comput. Phys., 306 (2016), pp. 1--18] and shown to have some favorable stability properties.