Regularization methods for the Poisson-Boltzmann equation: Comparison and accuracy recovery

A significant challenge in numerical solution to the Poisson-Boltzmann equation is due to singular charge sources in terms of Dirac delta functions. To overcome this difficulty, several regularization methods have been developed, in which the potential function is decomposed into two or three parts so that the singular component can be analytically solved using the Green's function, while other components become bounded. However, it was observed in the literature that some regularization methods are significantly less accurate than the others for unclear reasons, even though they are analytically equivalent. To understand this discrepancy, the numerical performance of four popular regularization methods is investigated in this work by implementing them with the Matched Interface and Boundary (MIB) approach, which is a sophisticated finite difference method for treating elliptic interface problems with discontinuous coefficients. With all four methods showing second order convergence, accuracy reduction is numerically observed in two schemes. This paper provides numerical analysis and experiment to trace the source of such reduction, and links the error to the fact that the Laplacian of Green's function is dropped outside the protein domain. While this term is analytically vanishing, its numerical negligence introduces a discretization error. Formulating via a proper elliptic interface problem, an effective accuracy recovery technique is proposed so that all four methods yield the same high precision. With this study, all involved regularization schemes are better understood and well connected into a unified framework.