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## Introduction

In this section, we shall discuss some simple numerical techniques for solving Poisson's equation: (108)

Here, is usually some sort of potential, whereas is a source term. The solution to the above equation is generally required in some simply-connected volume bounded by a closed surface . There are two main types of boundary conditions to Poisson's equation. In so-called Dirichlet boundary conditions, the potential is specified on the bounding surface . In so-called Neumann boundary conditions, the normal gradient of the potential 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 in terms of an electric potential : (109)

The potential itself satisfies Poisson's equation: (110)

where is the charge density, and the permittivity of free-space. In Newtonian gravity, we can write the force exerted on a unit test mass in terms of a gravitational potential : (111)

The potential satisfies Poisson's equation: (112)

where is the mass density, and the universal gravitational constant.   Next: 1-d problem with Dirichlet Up: Poisson's equation Previous: Poisson's equation
Richard Fitzpatrick 2006-03-29