On the Difference Method for Approximation of Second Order Derivatives of a Solution of Laplace’s Equation in a Rectangular Parallelepiped

Abstract

We present and justify finite difference schemes with the 14-point averaging operator for the second derivative of the solution of the Dirichlet problem for Laplace's equations on a rectangular parallelepiped. The boundary functions ϕ_{j} on the faces Γ_{j}, j=1,2,...,6 of the parallelepiped are supposed to have fifth derivatives belonging to the Hölder classes C^{5,λ}, 0<λ<1. 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.
It is proved that the proposed difference schemes for the approximation of the pure and mixed second derivatives converge uniformly with order O(h^{3+λ}), 0<λ<1 and O(h³), respectively.
Numerical experiments are illustrated to support the theoretical results.