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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 simplyconnected volume bounded by a
closed surface . There are two main types of boundary conditions to Poisson's equation.
In socalled Dirichlet boundary conditions, the potential is specified on the bounding
surface . In socalled 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
freespace. 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: 1d problem with Dirichlet
Up: Poisson's equation
Previous: Poisson's equation
Richard Fitzpatrick
20060329