Domain Decomposition Method for Poisson--Boltzmann Equations based on Solvent Excluded Surface

TL;DR

This paper introduces a novel domain decomposition method for solving the nonlinear Poisson-Boltzmann equation in computational chemistry, utilizing a hybrid solver and Schwarz decomposition to efficiently handle complex solvent-excluded surfaces.

Contribution

The paper presents a new domain decomposition approach with a hybrid linear-nonlinear solver for the Poisson-Boltzmann equation based on solvent-excluded surfaces.

Findings

The method effectively solves the nonlinear Poisson-Boltzmann equation.

Numerical experiments demonstrate the importance of the nonlinear model.

The approach improves computational efficiency for complex geometries.

Abstract

In this paper, we develop a domain decomposition method for the nonlinear Poisson-Boltzmann equation based on a solvent-excluded surface widely used in computational chemistry. The model relies on a nonlinear equation defined in $\mathbb{R}^3$ with a space-dependent dielectric permittivity and an ion-exclusion function that accounts for steric effects. Potential theory arguments transform the nonlinear equation into two coupled equations defined in a bounded domain. Then, the Schwarz decomposition method is used to formulate local problems by decomposing the cavity into overlapping balls and only solving a set of coupled sub-equations in each ball. The main novelty of the proposed method is the introduction of a hybrid linear-nonlinear solver used to solve the equation. A series of numerical experiments are presented to test the method and show the importance of the nonlinear model.