Accurate and fast numerical solution of Poisson's equation for arbitrary, space-filling Voronoi polyhedra: near-field corrections revisited

TL;DR

This paper introduces a fast, accurate numerical method for solving Poisson's equation in complex Voronoi polyhedra, effectively handling near-field corrections without convergence issues across various system types.

Contribution

The authors develop a novel, efficient approach that overcomes longstanding challenges in computing near-field corrections for arbitrary convex Voronoi polyhedra, applicable to diverse physical systems.

Findings

Method achieves O(NVP) computational complexity.

Avoids ill-convergent sums in near-field calculations.

Validated with numerical comparisons to exact models.

Abstract

We present an accurate and rapid solution of Poisson's equation for space-filling, arbitrarily- shaped, convex Voronoi polyhedra (VP); the method is O(NVP), where NVP is the number of distinct VP representing the system. In effect, we resolve the longstanding problem of fast but accurate numerical solution of the near-field corrections (NFC), contributions to each VP potential from nearby VP - typically involving multipole-type conditionally-convergent sums, or fast Fourier transforms. Our method avoids all ill-convergent sums, is simple, accurate, efficient, and works generally, i.e., for periodic solids, molecules, or systems with disorder or imperfections. We demonstrate the method's practicality by numerical calculations compared to exactly solvable models.