The Poisson equation

This chapter considers the Poisson equation. As with Laplace's equation, the Poisson equation is one of the important equations of mathematical physics, where it is commonly used to represent the potential field that results from some sink/source term. The Poisson equation is perhaps most familiar from its role in electrostatics, where it describes the electrical potential field that results from a known charge distribution. The chapter describes the domain of an example problem, which is set on a small, but well-known, Alcatraz island in the San Francisco Bay. A circular domain is especially attractive for modeling Alcatraz island. It is always appealing to reduce the dimensionality of a problem, because the cost increases roughly as the power of the number of dimensions. The chapter presents a nondimensionalization scheme that introduced the aspect ratio of the domain into the governing equation. It uses analytical and numerical methods to obtain an approximate solution to the example problem.