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 ϕ_{j} on the faces Γ_{j}, j=1,2,...,6 of the parallelepiped are supposed to have pth order derivatives satisfying the Hölder condition, i.e., ϕ_{j}∈C^{p,λ}(Γ_{j}), 0<λ<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 u_{h}-u of the approximate solution u_{h} at each grid point (x₁,x₂,x₃), |u_{h}-u|≤cρ^{p-4}(x₁,x₂,x₃)h⁴ is obtained, where u is the exact solution, ρ=ρ(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 ρ and h. It is proved that when ϕ_{j}∈C^{p,λ}, 0<λ<1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(h^{p-1}), p∈{4,5}.