A highly accurate homogeneous scheme for solving the laplace equation on a rectangular parallelepiped with boundary values in $C^{k,1}$

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace’s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is $O(h^4)$, where $h$ is the mesh step, when the boundary functions are from $C^{3,1}$, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is $O(h^6(|{\ln h}|+1))$, when the boundary functions are from $C^{5,1}$, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.