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.

翻译：本文提出并分析了两种用于数值求解泊松-能斯特-普朗克方程的稳定化有限元方法。我们所考虑的稳定化是通过对离子方程使用激波检测器和离散图拉普拉斯算子来实现的，而电势的离散方程则无需稳定化处理。由第一种算法得到的离散解同时保持离散极大值原理和极小值原理。对于第二种算法，其离散解被构造为在网格满足锐角条件时，能够保持离散极值原理并服从熵定律。值得注意的是，后者被证明是无条件稳定的。我们通过数值实验验证了所提方法的有效性。