A fast multi-resolution lattice Green's function method for elliptic difference equations

TL;DR

This paper introduces a multi-resolution lattice Green's function method that efficiently solves elliptic difference equations on unbounded domains by combining mesh refinement, fast multipole, and Fourier techniques, achieving high accuracy with reduced computational cost.

Contribution

It extends the FLGF method to a multi-resolution scheme with local mesh refinement, maintaining second-order accuracy and linear complexity while reducing degrees of freedom.

Findings

Achieves linear complexity in solving elliptic difference equations.

Maintains second-order accuracy with local mesh refinement.

Reduces computational cost significantly compared to uniform meshes.

Abstract

We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast multipole method in conjunction with fast Fourier transforms to yield linear complexity and decrease time-to-solution. We extend this method to a multi-resolution scheme and allow for locally refined Cartesian blocks embedded in the computational domain. Appropriately chosen interpolation and regularization operators retain consistency between the discrete Laplace operator and its inverse on the unbounded domain. Second-order accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost.