Accelerated cartesian harmonics for fast computation of time and frequency domain low-frequency kernels

Low frequency regime is defined as the domain wherein the spatial size is orders of magnitude smaller than the wavelength. Of course, in time domain, this translates to the smallest wavelength in the simulation. Capturing these geometric features necessitates dense discretization. Methods to overcome the bottlenecks posed by this problem have been an active research topic for a while, both in the time and frequency domain, see Refs. [J. Meng et al., 2006] and [J.S. Zhao and W.C. Chew, 2001] and references therein. The literature in frequency domain is considerably more extensive than time and the paper cited here is only representative of the work. In what follows, we present another scheme, that can be used for simulating low frequency problems in the frequency domain, and with very minor modifications, perform similar analysis in the time domain as well. The algorithms presented here rely on Accelerated Cartesian Expansions (ACE) [B. Shanker and H. He, 2006].