We present a second-order accurate algorithm for solving the free-space Poisson’s equation on a locally-refined nested grid hierarchy in three dimensions. Our approach is based on linear superposition of local convolutions of localized charge distributions, with the nonlocal coupling represented on coarser grids. The representation of the nonlocal coupling on the local solutions is based on Anderson’s Method of Local Corrections and does not require iteration between different resolutions. A distributed-memory parallel implementation of this method is observed to have a computational cost per grid point less than three times that of a standard FFT-based method on a uniform grid of the same resolution, and scales well up to 1024 processors.
"A local corrections algorithm for solving Poisson's equation in three dimensions." Commun. Appl. Math. Comput. Sci. 2 (1) 57 - 81, 2007. https://doi.org/10.2140/camcos.2007.2.57