A Self-Adjusting FEM-BEM Coupling Scheme for the Nonlinear Poisson-Boltzmann Equation

TL;DR

This paper introduces a self-adjusting FEM-BEM coupling method for solving the nonlinear Poisson-Boltzmann equation, enabling fast, reliable convergence without manual parameter tuning, validated on molecular structures.

Contribution

It presents an automatic relaxation parameter adjustment scheme within a FEM-BEM framework for nonlinear Poisson-Boltzmann equations, improving convergence and efficiency.

Findings

Newton-Raphson with cubic approximation is most effective

Optimal relaxation reduces iteration count by 40%

Achieved 1.37x speed-up over manual tuning

Abstract

The Poisson-Boltzmann equation is widely used to model molecular electrostatics; however, it is usually solved in linearised form because the sinh nonlinearity is challenging, limiting its applicability in highly charged systems such as nucleic acids. This work presents a solution method for the nonlinear Poisson-Boltzmann equation based on a coupled finite/boundary element scheme that automatically finds an optimal relaxation parameter, ensuring fast and reliable convergence of the nonlinear solver without user intervention. We validated our solver against APBS for a spherical cavity, and used RNA-based structures to perform a thorough study of the different algorithmic choices, and to test our implementation. We found that the best alternative to solve the Poisson-Boltzmann equation was using a Newton-Raphson method where the nonlinearity was gradually introduced with a cubic approximation in the first iteration. Newton-Raphson was also the best method to find the optimal relaxation factor, reducing the number of iterations by 40%. Including other optimisation techniques, we were able to obtain a 1.37x speed-up with respect to the best hand-picked relaxation factor for 1HC8 (molecule with highest charge in our tests), avoiding any trial-and-error process to find the relaxation factor.