# Laplace Equation¶

In electrostatics, the *Laplace equation* can calculate
the potentials throughout some volume of empty space
given certain known conditions on the boundary surface enclosing that volume.
Those boundary conditions are typically voltages on the surfaces of electrodes
(see *Dirichlet Boundary Conditions*) but can also be
planes of mirror symmetry (see *Neumann Boundary Condition*).

The Laplace equation is

for electric field, . It is simply a statement that the divergence of electric field is zero everywhere. It can also be written in terms of potential, (since ):

The Laplace equation is a special case of the *Poisson Equation* in
which zero space-charge is assumed throughout the volume (not including the
boundary surface, which is typically made up of electrode surfaces
having surface charges).

## SIMION Specific Notes¶

SIMION’s “Refine” function solves the Laplace equation in a *Potential Array*.
Electrode points in the PA are taken as Dirichlet boundary conditions with the given
potential, and non-electrode PA edges are taken as Neumann boundary conditions
(in most conditions–see *Neumann Boundary Condition*).