Convergence rate for numerical computation of the lattice Green's function

TL;DR

This paper analyzes the convergence rates of different numerical methods for computing the lattice Green's function, demonstrating that the discontinuity correction method is the most efficient for both isotropic and anisotropic cases.

Contribution

It provides a detailed comparison of three techniques for computing the lattice Green's function and confirms the superior efficiency of the discontinuity correction method.

Findings

Discontinuity correction method is most computationally efficient.

Convergence rates are calculated for isotropic and anisotropic cases.

Results match analytic predictions.

Abstract

Flexible boundary condition methods couple an isolated defect to bulk through the bulk lattice Green's function. The inversion of the force-constant matrix for the lattice Green's function requires Fourier techniques to project out the singular subspace, corresponding to uniform displacements and forces for the infinite lattice. Three different techniques--relative displacement, elastic Green's function, and discontinuity correction--have different computational complexity for a specified numerical error. We calculate the convergence rates for elastically isotropic and anisotropic cases and compare them to analytic results. Our results confirm that the discontinuity correction is the most computationally efficient method to compute the lattice Green's function.