Solving the Generalized Poisson Equation Using the Finite-Difference Method (FDM)

The Poisson equation is a very powerful tool for modeling the behavior of electrostatic systems, but unfortunately may only be solved analytically for very simplified models. Consequently, numerical simulation must be utilized in order to model the behavior of complex geometries with practical value. Although there are several competing algorithms for achieving this goal, one of the simplest and more straightforward of these is called the finite-difference method (FDM). At its core, FDM is nothing more than a direct conversion of the Poisson equation from continuous functions and operators into their discretely-sampled counterparts. This converts the entire problem into a system of linear equations that may be readily solved via matrix inversion. The accuracy of such a method is therefore directly tied to the ability of a finite grid to approximate a continuous system, and errors may be arbitrarily reduced by simply increasing the number of samples. Despite the relative simplicity of FDM as a numerical tool, information on the subject is surprisingly scarce. This is especially true for the case of quasi-static systems experiencing current flow in conducting materials. Part of the reason for this likely has to do with the fact that FDM is, at its core, nothing more than a simplified form of the finite element method (FEM). The only difference is that FDM is solved through a fixed, rectangular geometry, while FEM utilizes a flexible, triangular mesh. Nevertheless, the uniform grids inherent to FDM make it very intuitive to learn and to program, especially for students unfamiliar with techniques in numerical methods. Consequently, the learning curve for FEM is far steeper than it is for FDM, and often requires a whole semester of study to fully understand. On the other hand, expertise with FDM may be readily achieved in only a few weeks, and even serves as an intuitive springboard from which to study the more complex nature of FEM. The goal of this paper is to serve as a comprehensive introduction to the principles of FDM. Much of the basic information is readily found in standard textbooks [1, 2], though many of the practical details and advanced topics are difficult to find anywhere at all. This paper is therefore a compilation of knowledge based on my experience with FDM, as well as a primer on some of the more advanced topics that are almost nonexistent in the literature. The audience is specifically intended to include first-year students in computational electromagnetics, but more advanced professionals should still find useful reference material as well. The basic governing equations are derived directly from Maxwell’s equations and FDM is first introduced in its most basic formulation. The algorithm is then extended from the classical Poisson equation to the generalized Poisson equation in order to include the effects of varying dielectrics within the domain. Finally, we conclude with an extension