New Properties of 9-Point Finite Difference Solution of the Laplace Equation

Abstract

Two new properties of the 9-point finite difference solution of the Laplace equation are obtained, when the boundary functions are given from C (5,1). It is shown that the maximum error is of order O (h6 (vertical bar ln h vertical bar +1)), and this order cannot be obtained for the class of boundary functions from C (5,lambda), 0 < lambda < 1. These properties of the 9-point solution can be used to justify different versions of domain decomposition, composite grids, and combined methods.