We develop a numerical scheme for Poisson-Boltzmann-Nernst-Planck (PBNP) model. We adopt Gummel's method to treat the nonlinearity of PBNP where Poisson-Bolzmann equation and Nernst-Plank equation are iteratively solved, and then the idea of discontinuous bubble (DB) to solve the Poisson-Bolzmann equation is exploited [6]. First, we regularize the solution of Poisson-Bolzmann equation to remove the singularity. Next, we introduce the DB function as in [6] to treat the nonhomogeneous jump conditions of the regularized solution. Then, we discretize the discontinuous bubble and the bilinear form of Poisson-Bolzmann equation and solve the discretized linear problem by the immersed finite element method. Once Poisson-Bolzmann equation is solved, we apply the control volume method to solve Nernst-Plank equation via an upwinding concept. This process is repeated by updating the previous approximation until the total residual of the system decreases below some tolerance. We provide out numerical experiments. We observe optimal convergence rates for the concentration variable in all examples having analytic solutions. We observe that our scheme reflects well without oscillations the effect on the distribution of electrons caused by locating the singular charge close to the interface.

我们为Poisson-Boltzmann-Nernst-Planck（PBNP）模型开发了一个数值方案。我们采用Gummel方法处理PBNP的非线性问题，其中迭代求解了Poisson-Bolzmann方程和Nernst-Plank方程，然后利用非连续气泡（DB）的思想解决了Poisson-Bolzmann方程[6]。首先，我们对Poisson-Bolzmann方程的解进行正则化以消除奇点。接下来，我们引入[6]中的DB函数来处理正则化解的非齐次跳跃条件。然后，我们离散化了不连续气泡和Poisson-Bolzmann方程的双线性形式，并通过沉浸式有限元方法解决了离散化线性问题。一旦泊松-玻尔兹曼方程解出，我们将采用控制体积法通过迎风概念来求解能斯特-普朗克方程。通过更新先前的近似值来重复此过程，直到系统的总残差降低到某个容差以下。我们提供数值实验。我们在所有具有解析解的示例中观察到浓度变量的最佳收敛速度。我们观察到，我们的方案在没有振荡的情况下很好地反映了由于将奇异电荷定位在界面附近而对电子分布产生的影响。