Physics-based stabilized finite element approximations of the Poisson--Nernst--Planck equations

TL;DR

This paper introduces two stabilized finite element methods for solving the Poisson--Nernst--Planck equations, ensuring discrete principles and stability, validated through numerical experiments.

Contribution

The paper develops and analyzes two novel stabilized finite element algorithms for the Poisson--Nernst--Planck equations with improved stability and principle-preserving properties.

Findings

First algorithm preserves maximum and minimum principles.

Second algorithm is unconditionally stable under mesh acuteness.

Numerical experiments validate the effectiveness of the methods.

Abstract

We present and analyze two stabilized finite element methods for solving numerically the Poisson--Nernst--Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for the ion equations, whereas the discrete equation for the electric potential need not be stabilized. Discrete solutions stemmed from the first algorithm preserve both maximum and minimum discrete principles. For the second algorithm, its discrete solutions are conceived so that they hold discrete principles and obey an entropy law provided that an acuteness condition is imposed for meshes. Remarkably the latter is found to be unconditionally stable. We validate our methodology through numerical experiments.