Three-dimensional integrated Green Functions for the Poisson equation

The standard implementation of using FFTs to solve the Poisson equation with open boundary conditions on a Cartesian grid loses accuracy when the change in pG (the product of the charge density with the Green function) over a mesh cell becomes nonlinear; this is commonly encountered in high aspect ratio situations and results in poor efficiency due to the need for a very large number of grid points. A modification which solves this problem, the integrated Green function (IGF), has been implemented in two dimensions using linear basis functions and in three dimensions using constant basis functions. But, until recently, it has proved to be very difficult to implement IGF in three dimensions using linear basis functions. Recently significant progress has been made. We present both the implementation and test results for the three-dimensional extension.