The Shortley–Weller embedded finite-difference method for the 3D Poisson equation with mixed boundary conditions

This paper describes a method for the solution of the 3D Poisson equation, subject to mixed boundary conditions, on an irregularly shaped domain. A finite difference method is used, with the domain embedded in a rectangular grid. Quadratic treatment of the boundary conditions is shown to be necessary to obtain uniform error of O(Δ2)O(Δ2). This contrasts with the Dirichlet case where both quadratic and linear treatments give O(Δ2)O(Δ2) error, although the coefficient of error may be much larger for the linear case. Explicit error estimates demonstrating this behaviour are found for the 1D case with similar behaviour found in 2D and 3D numerical examples. Finally, the extension of this approach to the N  -dimensional case is given, where N>3N>3.