The quasi-2D electrostatic systems, characterized by periodicity in two dimensions with a free third dimension, have garnered significant interest in many fields. 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 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 case 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 its 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, achieving $O(N\log N)$ complexity with a small prefactor. The performance of the scheme is demonstrated via extensive numerical experiments.

翻译：准二维静电系统具有两个维度的周期性及第三个自由维度，在众多领域中引起了广泛关注。本文对拉普拉斯核应用高斯和近似，将相互作用分解为近场、中程和长程分量。近场分量具有奇异性但在局部区域内紧支撑，采用直接计算。中程分量通过类似于三维非均匀快速傅里叶变换的流程处理。长程分量包含大方差的高斯函数，在自由维度采用多项式插值/反插值技术，在另外两个维度通过代理点上的傅里叶谱求解器处理。与快速Ewald求和方法相比——该方法在高纵横比情况下需要大量零填充——高斯函数的可分离性使我们能够在自由方向上无需任何零填充即可处理此类情况。此外，虽然非均匀快速傅里叶变换通常需要在每个维度进行特定上采样，且截断核方法会因核振荡引入额外的上采样因子，但本方案得益于高斯函数的平滑性，无需在任何方向进行上采样，从而显著降低了大规模问题的计算成本。最后，尽管所有周期性快速多极子方法都需要将周期平铺划分为平滑的远场部分和包含最近邻单元的近场部分，但本方案直接在基本单元上操作，以更简单的实现获得了更优的性能。我们提供了严格的误差分析，证明在类非均匀快速傅里叶变换步骤中无需上采样，实现了具有小常数因子的$O(N\log N)$复杂度。通过大量数值实验验证了该方案的性能。