This work considers charged systems described by the modified Poisson--Nernst--Planck (PNP) equations, which incorporate ionic steric effects and the Born solvation energy for dielectric inhomogeneity. Solving the steady-state modified PNP equations poses numerical challenges due to the emergence of sharp boundary layers caused by small Debye lengths, particularly when local ionic concentrations reach saturation. To address this, we first reformulate the steady-state problem as a constraint optimization, where the ionic concentrations on unstructured Delaunay nodes are treated as fractional particles moving along edges between nodes. The electric fields are then updated to minimize the objective free energy while satisfying the discrete Gauss's law. We develop a local relaxation method on unstructured meshes that inherently respects the discrete Gauss's law, ensuring curl-free electric fields. Numerical analysis demonstrates that the optimal mass of the moving fractional particles guarantees the positivity of both ionic and solvent concentrations. Additionally, the free energy of the charged system consistently decreases during successive updates of ionic concentrations and electric fields. We conduct numerical tests to validate the expected numerical accuracy, positivity, free-energy dissipation, and robustness of our method in simulating charged systems with sharp boundary layers.

翻译：本文研究了由修正泊松-能斯特-普朗克（PNP）方程描述的带电系统，该方程考虑了离子空间位阻效应以及介电非均匀性下的玻恩溶剂化能。由于德拜长度极小导致的尖锐边界层的出现，特别是当局部离子浓度达到饱和时，求解稳态修正PNP方程面临数值挑战。为此，我们首先将稳态问题重构为约束优化问题，其中非结构化德劳内节点上的离子浓度被视作沿节点间边运动的分数粒子。随后更新电场以最小化目标自由能，同时满足离散高斯定律。我们开发了一种非结构化网格上的局部松弛方法，该方法内蕴地满足离散高斯定律，确保电场无旋。数值分析表明，移动分数粒子的最优质量保证了离子浓度和溶剂浓度的正定性。此外，在离子浓度和电场的连续更新过程中，带电系统的自由能持续降低。我们进行了数值测试，验证了所提方法在模拟具有尖锐边界层的带电系统时的预期数值精度、正定性、自由能耗散特性及鲁棒性。