FFT BASED SPECTRAL EWALD METHODS AS AN ALTERNATIVE TO FAST MULTIPOLE METHODS

In this paper, we review a set of fast and spectrally accurate methods for rapid evaluation of three dimensional electrostatic and Stokes potentials. The algorithms use the so-called Ewald decomposition and are FFT-based, which makes them naturally most efficient for the triply periodic case. Two key ideas have allowed efficient extension of these Spectral Ewald (SE) methods to problems with periodicity in only one or two dimensions: an adaptive 3D FFT that apply different upsampling rates locally combined with a newmethod for FFT based solutions of free space harmonic and biharmonic problems. The latter approach is also used to extend to the free space case, with no periodicity. For the non-radial kernels of Stokes flow, the structure of their Fourier transform is exploited to extend the applicability from the radial harmonic and biharmonic kernels. Awindow function is convolvedwith the point charges to assign values on the FTT grid. Spectral accuracy is attained with a variable number of points in the support of the window function, tuning a shape parameter according to this choice. A new window function, recently introduced in the context of a non-uniform FFT algorithm, allows for further reduction in the computational time as compared to the truncated Gaussians previously used in the SE method.