  The efficient simulation of fluid-structure interactions at zero Reynolds
number requires the use of fast summation techniques in order to rapidly
compute the long-ranged hydrodynamic interactions between the structures. One
approach for periodic domains involves utilising a compact or exponentially
decaying kernel function to spread the force on the structure to a regular grid
where the resulting flow and interactions can be computed efficiently using an
FFT-based solver. A limitation to this approach is that the grid spacing must
be chosen to resolve the kernel and thus, these methods can become inefficient
when the separation between the structures is large compared to the kernel
width. In this paper, we address this issue for the force-coupling method (FCM)
by introducing a modified kernel that can be resolved on a much coarser grid,
and subsequently correcting the resulting interactions in a pairwise fashion.
The modified kernel is constructed to ensure rapid convergence to the exact
hydrodynamic interactions and a positive-splitting of the associated mobility
matrix. We provide a detailed computational study of the methodology and
establish the optimal choice of the modified kernel width, which we show plays
a similar role to the splitting parameter in Ewald summation. Finally, we
perform example simulations of rod sedimentation and active filament
coordination to demonstrate the performance of fast FCM in application.
