Single cell high order difference schemes for Poisson's equation in three variables

In this article we present high order difference schemes for the Poisson's equation in three variables with Dirichlet or Neumann boundary conditions. Formulae of order two, four and six are derived on a single cubic cell of size 2h. The procedure is also extended to derive difference schemes of order two and four for a similar equation with variable coefficients. The resulting system of algebraic equations could be solved by standard iterative methods. Numerical results of some test problems demonstrating the effectiveness of the difference schemes are appended.