O(h8 |ln h|) order of accurate difference method for solving the Dirichlet problem for Laplace’s equation on a rectangle with boundary values in Ck,1

A three stage (9-point, 5-point and 5-point) difference method for solving the Dirichlet problem for Laplace's equation on a rectangle is proposed and justified. It is proved that the proposed difference solution converges uniformly to the exact solution of order O(h8 |ln h|), h is the mesh size, when the boundary functions are from C9,1. Numerical experiment is illustrated to support the analysis made.