Bowen WR, Sharif AO
Department of Chemical Engineering, University of Wales Swansea, Swansea, SA2 8PP, United Kingdom
J Colloid Interface Sci. 1997 Mar 15;187(2):363-74. doi: 10.1006/jcis.1996.4705.
A Galerkin finite-element approach combined with an error estimator and automatic mesh refinement has been used to provide a flexible numerical solution of the Poisson-Boltzmann equation. A Newton sequence technique was used to solve the nonlinear equations arising from the finite-element discretization procedure. Errors arising from the finite-element solution due to mesh refinement were calculated using the Zienkiewicz-Zhu error estimator, and an automatic remeshing strategy was adopted to achieve a solution satisfying a preset quality. Examples of the performance of the error estimator in adaptive mesh refinement are presented. The adaptive finite-element scheme presented in this study has proved to be an effective technique in minimizing errors in finite-element solutions for a given problem, in particular those of complex geometries. As an example, numerical solutions are presented for the case of a charged spherical particle at various distances from a charged cylindrical pore in a charged planar surface. Such a scheme provides a quantification of the significance of electrostatic interactions for an important industrial technology-membrane separation processes.
一种结合误差估计器和自动网格细化的伽辽金有限元方法已被用于提供泊松-玻尔兹曼方程的灵活数值解。使用牛顿序列技术来求解有限元离散化过程中产生的非线性方程。利用齐恩凯维奇-朱误差估计器计算由于网格细化导致的有限元解中的误差，并采用自动重新网格化策略来获得满足预设质量的解。给出了误差估计器在自适应网格细化中的性能示例。本研究中提出的自适应有限元方案已被证明是一种有效的技术，可将给定问题的有限元解中的误差降至最低，特别是对于复杂几何形状的问题。例如，给出了带电球形颗粒在带电平面表面上与带电圆柱形孔不同距离处的数值解。这种方案为重要的工业技术——膜分离过程中静电相互作用的重要性提供了量化。
