Fast Ewald summation for Green’s functions of Stokes flow in a half-space

Abstract

Recently, Gimbutas et al in [1] derived an elegant representation for the Green's functions of Stokes flow in a half-space. We present a fast summation method for sums involving these halfspace Green's functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al [2] for the corresponding free-space Green's functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real-space (short-range interactions) and one in Fourier-space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier-space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison to the freespace evaluation, greater computational savings is also achieved when compared to their respective direct sums. 0 conditionally convergent sum into two rapidly converging sums -one in real space and one in Fourier space. The computational complexity is however quadratic in the number of points. The survey by Deserno and Holm [7] traces the development of fast methods based on FFTs for acceleration of the Fourier space sum in the context of electrostatics and molecular dynamics. An early method called Smooth Particle Mesh Ewald (SPME) [8] was utilized by Saintillan et al [9] for fast evaluation of periodic stokeslet sums. Later, Tornberg et al developed a Spectral Ewald (SE) method for the 3periodic sum of stokeslets [10], stresslets [11] and rotlets [12] that is spectrally accurate and recovers the exponentially fast convergence of the Ewald sums that traditional Particle Mesh Ewald approaches cannot.The SE method is best suited for the 3-periodic case. Otherwise, for every direction that is not periodic, oversampling of the FFTs becomes necessary to compute the aperiodic convolution, and this increases the computational cost. The work in [13] illustrates the use of the SE method for a 2-periodic sum of stokeslets, while in [14] the SE method is adapted for 1-periodic sums in the context of electrostatics. The case of free-space sum of stokeslets (no periodicity) is the most challenging for the SE method and it was solved recently by Klinteberg et al [2] by combining two different ideas. The first idea is the free-space solution of harmonic and biharmonic equations using FFT on a uniform grid by Vico et al [15] that amounts to the convolution of harmonic/biharmonic (radial) kernels with source terms by FFT on a uniform grid, and the second idea is that the stokeslet, stresslet and rotlet kernels, though not radial, can be expressed as a linear combination of differential operations on the harmonic or biharmonic kernels.A popular method ideally suited for free-space problems is t...

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