On a highly accurate approximation of the first and pure second derivatives of the Laplace equation in a rectangular parallelpiped

We propose and justify difference schemes for the approximation of the first and pure second derivatives of a solution of the Dirichlet problem in a rectangular parallelepiped. The boundary values on the faces of the parallelepiped are supposed to have six derivatives satisfying the Holder condition, to be continuous on the edges, and to have second-and fourth-order derivatives satisfying the compatibility conditions resulting from the Laplace equation. We prove that the solutions of the proposed difference schemes converge uniformly on the cubic grid of order O(h(4)), where h is a grid step. Numerical experiments are presented to illustrate and support the analysis made.