A modified combined grid method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A modified combined grid method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. The six-point averaging operator is applied at next-to-the-boundary grid points, while the 18-point averaging operator is used instead of the 26-point one at the remaining grid points. Assuming that the boundary values given on the faces have fourth derivatives satisfying the Hölder condition, the boundary values on the edges are continuous, and their second derivatives obey a matching condition implied by the Laplace equation, the grid solution is proved to converge uniformly with the fourth order with respect to the mesh size.