Fast summation of Stokes potentials using a new kernel-splitting in the DMK framework

Abstract

Classical Ewald methods for Coulomb and Stokes interactions rely on ``kernel-splitting," using decompositions based on Gaussians to divide the resulting potential into a near field and a far field component. Here, we show that a more efficient splitting for the scalar biharmonic Green's function can be derived using zeroth-order prolate spheroidal wave functions (PSWFs), which in turn yields new efficient splittings for the Stokeslet, stresslet, and elastic kernels, since these Green's tensors can all be derived from the biharmonic kernel. This benefits all fast summation methods based on kernel splitting, including FFT-based Ewald summation methods, that are suitable for uniform point distributions, and DMK-based methods that allow for nonuniform point distributions. The DMK (dual-space multilevel kernel-splitting) algorithm we develop here is fast, adaptive, and linear-scaling, both in free space and in a periodic cube. We demonstrate its performance with numerical examples in two and three dimensions.