Introduction

In this section, we shall discuss some simple numerical techniques for solving Poisson's equation:

$$\nabla^2 u({\bf r}) = v({\bf r}).$$ (108)

Here, $u({\bf r})$ is usually some sort of potential, whereas $v({\bf r})$ is a source term. The solution to the above equation is generally required in some simply-connected volume $V$ bounded by a closed surface $S$. There are two main types of boundary conditions to Poisson's equation. In so-called Dirichlet boundary conditions, the potential $u$ is specified on the bounding surface $S$. In so-called Neumann boundary conditions, the normal gradient of the potential $\nabla u\cdot d{\bf S}$ is specified on the bounding surface.

Poisson's equation is of particular importance in electrostatics and Newtonian gravity. In electrostatics, we can write the electric field ${\bf E}$ in terms of an electric potential $\phi$:

$${\bf E} = - \nabla\,\phi.$$ (109)

The potential itself satisfies Poisson's equation:

$$\nabla^2 \phi = - \frac{\rho}{\epsilon_0},$$ (110)

where $\rho({\bf r})$ is the charge density, and $\epsilon _0$ the permittivity of free-space. In Newtonian gravity, we can write the force ${\bf f}$ exerted on a unit test mass in terms of a gravitational potential $\phi$:

$${\bf f} = - \nabla\,\phi.$$ (111)

The potential satisfies Poisson's equation:

$$\nabla^2 \phi = 4\pi^2\,G\,\rho,$$ (112)

where $\rho({\bf r})$ is the mass density, and $G$ the universal gravitational constant.