Numerical Solutions to Poisson Equations Using the Finite-Difference Method

The Poisson equation is an elliptic partial differential equation that frequently emerges when modeling electromagnetic systems. However, like many other partial differential equations, exact solutions are difficult to obtain for complex geometries. This motivates the use of numerical methods in order to provide accurate results for real-world systems. One very simple algorithm is the Finite-Difference Method (FDM), which works by replacing the continuous derivative operators with approximate finite differences. Although the Finite-Difference Method is one of the oldest methods ever devised, comprehensive information is difficult to find compiled in a single reference. This paper therefore provides a tutorial-level derivation of the Finite-Difference Method from the Poisson equation, with special attention given to practical applications such as multiple dielectrics, conductive materials, and magnetostatics.