Fast summation on rectangular cuboids with arbitrary periodicity in the DMK framework

Dual-space multilevel kernel-splitting (DMK) is a fast summation framework that combines ideas from the fast multipole method, Ewald summation, and multilevel summation.

Originally formulated for free-space problems, and later extended to fully periodic problems on a cube, it decomposes the kernel interaction into a smooth global contribution and a hierarchy of localized interactions evaluated on an octree.

We extend DMK to problems on rectangular cuboids with periodic boundary conditions in one, two, or three coordinate directions.

The periodization leverages the fact that interactions on all tree levels below the root are localized, allowing for their evaluation with minimal modification on a cubical tiling of the domain.

The remaining smooth root-level far-field contribution is evaluated in Fourier space, with Fourier series in the periodic directions and Fourier integrals in the free directions.

For reduced periodicity, truncated kernels are used to regularize singular and near-singular Fourier kernels, yielding rapidly convergent trapezoidal discretizations and a unified treatment of all periodicities.

For large-aspect-ratio cuboids, the root-level sum can be accelerated using the fast Fourier transform.

We validate the method for the electrostatic potential and Stokeslet, stresslet and rotlet potentials, for all periodicities and a wide range of aspect ratios.

Numerical experiments show that the periodization adds only a small overhead to the original free-space DMK algorithm, also for high-aspect-ratio cuboids.

The resulting method provides a framework for applying DMK to problems with mixed periodicity on rectangular cuboids, and extends naturally to other non-oscillatory kernels for which a kernel split is available.