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Introduction

In this section, we shall discuss some simple numerical techniques for solving Poisson's equation:
\begin{displaymath}
\nabla^2 u({\bf r}) = v({\bf r}).
\end{displaymath} (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 $\phi$:

\begin{displaymath}
{\bf E} = - \nabla\,\phi.
\end{displaymath} (109)

The potential itself satisfies Poisson's equation:
\begin{displaymath}
\nabla^2 \phi = - \frac{\rho}{\epsilon_0},
\end{displaymath} (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 $\phi$:
\begin{displaymath}
{\bf f} = - \nabla\,\phi.
\end{displaymath} (111)

The potential satisfies Poisson's equation:
\begin{displaymath}
\nabla^2 \phi = 4\pi^2\,G\,\rho,
\end{displaymath} (112)

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


next up previous
Next: 1-d problem with Dirichlet Up: Poisson's equation Previous: Poisson's equation
Richard Fitzpatrick 2006-03-29