A general meshsize fourth-order compact difference discretization scheme for 3D Poisson equation

A fourth-order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three-dimensional Poisson equation on a regular cubic domain. The difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three-dimensional manipulations. We use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth-order finite difference scheme.