A simple form for the fourth order difference method for 3-D elliptic equations

In 1992, Jain et al. [M.K. Jain, R.K. Jain, R.K. Mohanty, Fourth-order finite difference method for three dimensional elliptic equations with nonlinear first-derivative terms, Numer. Meth. Part. Differ. Equat. 8 (1992) 575-591] proposed a fourth order finite difference scheme for the 3-D elliptic equation. In this paper, we present a simple and new form of 19-point fourth order difference method for the nonlinear second-order 3-D elliptic difference equation Auxx + Buyy + Cuzz = f(x, y, z, u, ux, uy, uz), where A, B and C are constants on a cubic region W subject to the Dirichlet boundary conditions on ∂W.