On the convergence in $C^1_h$ of the difference solution to the Laplace equation in a rectangular parallelepiped

The Dirichlet problem for the Laplace equation in a rectangular parallelepiped is considered. It is assumed that the boundary values have the third derivatives on the faces that satisfy the Hцlder condition, the boundary values are continuous on the edges, and their second derivatives satisfy the compatibility condition that is implied by the Laplace equation. The uniform convergence of the grid solution of the Dirichlet problem and of its difference derivative on the cubic grid at the rate $O(h^2)$, where $h$ is the grid size, is proved. A piecewise polylinear continuation of the grid solution and of its difference derivative uniformly approximate the solution of the Dirichlet problem and its second derivative on the close parallelepiped with the second order of accuracy with respect to $h$.