On the Fourth-Order Accurate Approximations of the Solution of the Dirichlet Problem for Laplace's Equation in a Rectangular Parallelepiped

An interpolation operator is proposed using the cubic grid solution of order 0 (h(4)), h is the mesh size, of the Dirichlet problem for Laplace's equation in a rectangular paralellepiped. It is proved that when the boundary functions on the faces of the rectangular parallelepiped are from the Holder classes C-4,C-lambda, lambda is an element of (0, 1), and their second and fourth derivatives obey compatibility conditions implied by Laplace's equation on the edges, the solution obtained by the constructed operator also has fourth-order accuracy with respect to mesh size