Poisson problem

Consider the Poisson equation

\[\nabla^2 \phi = f\]

This is a second-order elliptic equation, and therefore requires 2 boundary conditions.

There is no time-dependence in this equation. The potential \(\phi\) responds instantaneously to the source \(f\) and the boundary conditions.

Consider the one-dimensional Poisson equation:

\[\phi^{\prime\prime} = f\]

on the domain \([a, b]\).

At each boundary. we can specify the boundary conditions as

• Dirichlet: \(\phi(a) = A\)

• Neumann: \(\phi^\prime(a) = C\)

• Robin: \(\alpha \phi(a) + \beta \phi^\prime(a) = \gamma\) (a mix of the above two)

• periodic

We can also have different boundary conditions on each boundary. If the values are set to 0, we call the conditions homogeneous, otherwise we call them inhomogeneous.

Note

Not any set of boundary conditions is allowed. Consider \(f = 0\), so our Poisson equation (is the Laplace equation):

\[\phi^{\prime\prime} = 0\]

and the solution is just

\[\phi(x) = a x + b\]

if we try to enforce different inhomogeneous Neumann boundary conditions on each end, then we get conflicting values for the slope—this is unsolvable.

To understand solvable boundary conditions for the general case,

\[\nabla^2 \phi = f\]

we start by integrating over the domain

\[\int_\Omega \nabla^2 \phi d\Omega = \int_{\partial \Omega} \nabla \phi \cdot {\bf n} dS = \int_\Omega f d\Omega\]

If we have homogeneous Neumann BCs on all sides, \(\nabla \phi \cdot {\bf n} = 0\), then the source, \(f\), must satisfy

\[\int_\Omega f d\Omega = 0\]

The same condition will apply, if the boundary conditions are periodic.