Flexibly imposing periodicity in kernel independent FMM: A Multipole-To-Local operator approach
This work addresses the need for flexible and efficient periodic boundary conditions in KIFMM, which is important for simulations in domains like electrostatics and fluid dynamics.
The paper presents a method to impose periodic boundary conditions in the kernel independent fast multipole method (KIFMM) using a near-far splitting scheme, where the far-field contribution is computed via a multipole-to-local operator. The method achieves O(N) complexity and maintains the same error bounds as KIFMM, with accuracy and timing tests shown for Laplace and Stokes kernels.
An important but missing component in the application of the kernel independent fast multipole method (KIFMM) is the capability for flexibly and efficiently imposing singly, doubly, and triply periodic boundary conditions. In most popular packages such periodicities are imposed with the hierarchical repetition of periodic boxes, which may give an incorrect answer due to the conditional convergence of some kernel sums. Here we present an efficient method to properly impose periodic boundary conditions using a near-far splitting scheme. The near-field contribution is directly calculated with the KIFMM method, while the far-field contribution is calculated with a multipole-to-local (M2L) operator which is independent of the source and target point distribution. The M2L operator is constructed with the far-field portion of the kernel function to generate the far-field contribution with the downward equivalent source points in KIFMM. This method guarantees the sum of the near-field \& far-field converge pointwise to results satisfying periodicity and compatibility conditions. The computational cost of the far-field calculation observes the same $\mathcal{O}(N)$ complexity as FMM and is designed to be small by reusing the data computed by KIFMM for the near-field. The far-field calculations require no additional control parameters, and observes the same theoretical error bound as KIFMM. We present accuracy and timing test results for the Laplace kernel in singly periodic domains and the Stokes velocity kernel in doubly and triply periodic domains.