Interval nine-point finite difference method for solving the laplace equation with the dirichlet boundary conditions

Abstract

An interval version of the conventional nine-point finite difference method for solving the two-dimensional Laplace equation with the Dirichlet boundary conditions is proposed. This interval scheme is interesting due to the fact that the local truncation error of the conventional method is of the high (fourth) order, but it becomes of the sixth order for square mesh. In the theoretical approach presented, this error is bounded by some interval values and we can prove that the exact solution belongs to the interval solutions obtained.