A reduced basis method for the nonlinear Poisson-Boltzmann equation
It provides a computationally efficient solution for repeated simulations of charged micro/nano scale systems, addressing a bottleneck in computational physics.
The paper adapts the reduced basis method (RBM) to efficiently solve the nonlinear Poisson-Boltzmann equation, achieving high accuracy and reliability in error estimation while capturing boundary layers in numerical tests.
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.