Efficient solution of Poisson's equation with free boundary conditions

TL;DR

This paper presents an efficient Fourier-based method using interpolating scaling functions to accurately solve Poisson's equation with free boundary conditions, achieving computational complexity comparable to periodic cases.

Contribution

It introduces a fast Fourier method with interpolating scaling functions for free boundary conditions, matching the efficiency of plane wave methods for periodic problems.

Findings

Achieves O(N log N) computational complexity.

Provides highly accurate electrostatic potentials.

Treats free boundary conditions as efficiently as periodic ones.

Abstract

Interpolating scaling functions give a faithful representation of a localized charge distribution by its values on a grid. For such charge distributions, using a Fast Fourier method, we obtain highly accurate electrostatic potentials for free boundary conditions at the cost of O(N log N) operations, where N is the number of grid points. Thus, with our approach, free boundary conditions are treated as efficiently as the periodic conditions via plane wave methods.