Fast Ewald summation for free-space Stokes potentials

TL;DR

This paper introduces a spectrally accurate, fast Fourier transform-based method for efficiently computing free-space Stokes potentials, achieving O(N log N) complexity and competitive performance with FMM.

Contribution

It adapts the Spectral Ewald framework to free-space Stokes potentials using FFTs, enabling rapid and accurate evaluations with minimal oversampling.

Findings

Achieves spectral accuracy in free-space Stokes potential computations.

Computational complexity of O(N log N) for large N.

Performance comparable to fast multipole methods.

Abstract

We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e. sums involving a large number of free space Green's functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems, with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid. Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of O(N log N) for problems with N sources and targets. Comparison is made with a fast multipole method (FMM) to show that the performance of the new method is competitive.