A simple form for the fourth order difference method for 3-D elliptic equations - Dehghan, Mehdi; Molavi-Arabshahi, Seyedeh Mahboubeh (Applied mathematics and computation, 2007)
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 Au xx + Bu yy + Cu zz = f( x, y, z, u, u x , u y , u z ), where A, B and C are constants on a cubic region W subject to the Dirichlet boundary conditions on ∂ W.