A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation

A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation

Year:    2019

Author:    Lijie Ji, Yanlai Chen, Zhenli Xu

Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 5 : pp. 1200–1218

Abstract

In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated resolution 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 high-fidelity numerical solutions of the Poisson-Boltzmann equation upon which the fast RBM 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2018-0188

Advances in Applied Mathematics and Mechanics, Vol. 11 (2019), Iss. 5 : pp. 1200–1218

Published online:    2019-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Reduced order modeling reduced basis method Poisson-Boltzmann equation differential capacitance.

Author Details

Lijie Ji

Yanlai Chen

Zhenli Xu