On the fourth order accurate interpolation operator for the difference solution of the 3-dimensional Laplace equation

A three-dimensional (3D) matching operator is proposed for a fourth-order accurate solution of a Dirichlet problem of Laplace's equation in a rectangular parallelepiped. The operator is constructed based on homogeneous, orthogonal-harmonic polynomials in three variables, and employs a cubic grid difference solution of the problem for the approximate solution inbetween the grid nodes. The difference solution on the nodes used by the interpolation operator is calculated by a novel formula, developed on the basis of the discrete Fourier transform. This formula can be applied on the required nodes directly, without requiring the solution of the whole system of difference equations. The fourth-order accuracy of the constructed numerical tools is demonstrated further through a numerical example.