On the order of maximum error of the finite difference solutions of Laplace's equation on rectangles

Abstract

The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C_{1,1} is analized. It is shown that the maximum error for 9-point approximation is of order O(h²(|ln h|+1)) as 5-point approximation. This order can be improved up to O(h²) when the 9-point approximation in the grids which are in h distance from the boundary is replaced by 5-point approximation (¨5 and 9¨-point scheme ). It is also proved that the class of boundary functions C_{1,1} to get the obtained error estimations cannot be essentially enlarged. Numerical experiments are illustrated to support the analysis made. These results point at the importance of taking the smothness of the boundary functions into account in choosing the numerical algorithms in applied problems. doi:10.1017/S1446181108000151