Convergence of a diffuse interface Poisson-Boltzmann (PB) model to the sharp interface PB model: A unified regularization formulation

Abstract

Both the sharp interface and diffuse interface Poisson-Boltzmann (PB) models have been developed in the literature for studying electrostatic interaction between a solute molecule and its surrounding solvent environment. In the mathematical analysis and numerical computation for these PB models, a significant challenge is due to singular charge sources in terms of Dirac delta distributions. Recently, based on various regularization schemes for the sharp interface PB equation, the first regularization method for the diffuse interface PB model has been developed in [S. Wang, E. Alexov, and S. Zhao, Mathematical Biosciences and Engineering, 18, 1370–1405, (2021)] for analytically treating the singular charges. This work concerns with the convergence of a diffuse interface PB model to the sharp interface PB model, as the diffused Gaussian-convolution surface (GCS) approaches to the sharp solvent accessible surface (SAS). Due to the limitation in numerical algorithm and mesh resolution, such a convergence is impossible to be verified numerically. Through analyzing the weak solution for the regularized PB equations, the convergences for both the reaction-field potential and electrostatic free energy are rigorously proved in this work. Moreover, this study provides a unified regularization for both sharp interface and diffuse interface PB models, and clarifies the connection between this unified formulation and the existing regularizations. This lays a theoretical foundation to develop regularization for more complicated PB models.