Accuracy analysis of a spectral Poisson solver

We solve Poissonʹs equation in d = 2 , 3 space dimensions by using a spectral method based on Fourier decomposition. The choice of the basis implies that Dirichlet boundary conditions on a box are satisfied. A Greenʹs function-based procedure allows us to impose Dirichlet conditions on any smooth closed boundary, by doubling the computational complexity. The error introduced by the spectral truncation and the discretization of the charge distribution is evaluated by comparison with the exact solution, known in the case of elliptical symmetry. To this end boundary conditions on an equipotential ellipse (ellipsoid) are imposed on the numerical solution. Scaling laws for the error dependence on the number K of Fourier components for each space dimension and the number N of point charges used to simulate the charge distribution are presented and tested. A procedure to increase the accuracy of the method in the beam core region is briefly outlined.