Volume 11, Issue 5
A Reduced Basis Method for the Nonlinear Poisson-Boltzmann Equation

Lijie Ji ,  Yanlai Chen and Zhenli Xu

10.4208/aamm.OA-2018-0188

Adv. Appl. Math. Mech., 11 (2019), pp. 1200-1218.

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  • 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.

  • History

Published online: 2019-06

  • AMS Subject Headings

65M10, 78A48

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