A Highly Accurate Difference Method for Solving the Dirichlet Problem for Laplace's Equation on a Rectangle

O(h(8)) order (h is the mesh size) of accurate three-stage difference method on a square grid for the approximate solution of the Dirichlet problem for Laplace's equation on a rectangle is proposed and justified without taking more than 9 nodes of the grid. At the first stage, by using the 9-point scheme the sum of the pure fourth derivatives of the desired solution is approximated of order O(h(6)). At the second stage, approximate values of the sum of the pure eighth derivatives is approximated of order O(h(2)) by the 5-point scheme. At the final third stage, the system of simplest 5-point difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first and second stages. Numerical experiment is illustrated to support the analysis made.