14-point difference operator for the approximation of the first derivatives of a solution of Laplace’s equation in a rectangular parallelepiped

Abstract

A 14-point difference operator is used to construct finite difference problems for the approximation of the solution, and the first order derivatives of the Dirichlet problem for Laplace?s equations in a rectangular parallelepiped. The boundary functions ?j on the faces ?j, j = 1,2,...,6 of the parallelepiped are supposed to have pth order derivatives satisfying the H?lder condition, i.e., ?j ? Cp,?(?j), 0 < ? < 1, where p = {4,5}. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh - u of the approximate solution uh at each grid point (x1,x2,x3), ?uh-u?? c?p-4(x1,x2,x3)h4 is obtained, where u is the exact solution, ? = ? (x1, x2,x3) is the distance from the current grid point to the boundary of the parallelepiped, h is the grid step, and c is a constant independent of ? and h. It is proved that when ?j ? Cp,?, 0 < ? < 1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(hp-1), p ? {4,5}.