An efficient method for the approximate solution of the Laplace difference equation on rectangular domains

An approximate method of solving Laplace's difference equation in a rectangle with uniform accuracy $O(h^p)$ is proposed. Five-point and nine-point Laplace difference operators, boundary conditions of the first and second kinds, and the problem of the contact between two media are considered. The method can be generalized to the three-dimensional case. To find an approximate solution, $O(1)$ arithmetic operations are required for each node of the net.