Numerical Solution of the Poisson Equation Using Finite Difference Matrix Operators

The Poisson equation frequently emerges in many fields of science and engineering. As exact solutions are rarely possible, numerical approaches are of great interest. Despite this, a succinct discussion of a systematic approach to constructing a flexible and general numerical Poisson solver can be difficult to find. In this introductory paper, a comprehensive discussion is presented on how to build a finite difference matrix solver that can solve the Poisson equation for arbitrary geometry and boundary conditions. The boundary conditions are implemented in a systematic way that enables easy modification of the solver for different problems. An image-based geometry-definition approach is also discussed. Python code of the numerical recipe is made publicly available. Numerical examples are presented that show how to set up the solver for different problems.