A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems

XZ Gao and SD Jiang and JY Liang and ZL Xu and Q Zhou, NUMERISCHE MATHEMATIK (2025).

DOI: 10.1007/s00211-025-01518-y

The quasi-2D electrostatic systems, characterized by periodicity in two dimensions with a free third dimension, have garnered significant interest in the fields of semiconductor physics, new energy technologies, and nanomaterials. We apply the sum-of-Gaussians (SOG) approximation to the Laplace kernel, dividing the interactions into near-field, mid-range, and long-range components. The near-field component, singular but compactly supported in a local domain, is directly calculated. The mid-range component is managed using a procedure similar to nonuniform fast Fourier transforms (NUFFTs) in three dimensions. The long-range component, which includes Gaussians of large variance, is treated with polynomial interpolation/anterpolation in the free dimension and Fourier spectral solver in the other two dimensions on proxy points. Unlike the fast Ewald summation, which requires extensive zero padding in the case of high aspect ratios, the separability of Gaussians allows us to handle such cases without any zero padding in the free direction. Furthermore, while NUFFTs typically rely on certain upsampling in each dimension, and the truncated kernel method introduces an additional factor of upsampling due to kernel oscillation, our scheme eliminates the need for upsampling in any direction due to the smoothness of Gaussians, significantly reducing computational cost for large-scale problems. Finally, whereas all periodic fast multipole methods require dividing the periodic tiling into a smooth far part and a near part containing the nearest neighboring cells, our scheme operates directly on the fundamental cell, resulting in better performance with simpler implementation. We provide a rigorous error analysis showing that upsampling is not required in NUFFT-like steps, and develop a careful parameter selection scheme to balance various parts of the whole scheme, achieving O(NlogN)\documentclass12ptminimal \usepackageamsmath \usepackagewasysym \usepackageamsfonts \usepackageamssymb \usepackageamsbsy \usepackagemathrsfs \usepackageupgreek \setlength\oddsidemargin-69pt \begindocument$$O(N\log N)$$\enddocument complexity with a small prefactor. The performance of the scheme is demonstrated via extensive numerical experiments. The scheme can be readily extended to deal with many other kernels, owing to the general applicability of the SOG approximation.

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