A reduced basis method for the nonlinear Poisson-Boltzmann equation

TL;DR

This paper introduces a novel application of the reduced basis method to efficiently solve the nonlinear Poisson-Boltzmann equation, significantly reducing computational costs while maintaining accuracy in complex charged systems.

Contribution

It adapts a rigorous model order reduction technique, the reduced basis method, to the nonlinear Poisson-Boltzmann equation for the first time, enabling faster simulations.

Findings

High efficiency and accuracy of the reduced basis algorithm

Reliable error estimation demonstrated

Effective boundary layer capturing

Abstract

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the {\em truth approximations} of the RBM upon which the fast algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.