Fast Poisson solver in a three-dimensional ellipsoid

Abstract

We present a simple and efficient fast direct solver for 3D Poisson problem in elliptical coordinates. The method relies on representing the solution as a truncated Fourier series, then solving the partial differential equation of Fourier coefficients by second-order finite difference discretizations. Using a grid by shifting half mesh away from the poles and incorporating the derived numerical boundary values, the difficulty of coordinate singularities can be elevated easily. Besides, the resulting linear equations can be solved very efficiently by some existing numerical algorithms.