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import numpy as np
import matplotlib.pyplot as plt

Poisson problem

Consider the Poisson equation

$${\mathrm{\nabla }}^2\varphi =f$$

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

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

Consider the one-dimensional Poisson equation:

$${\varphi }^{\prime \prime }=f$$

on the domain $[a,b]$.

We can supply boundary conditions as

• Dirichlet: $\varphi (a)=A$

• Neumann: ${\varphi }^{\prime }(a)=C$

or a mix of the two. If the values are set to 0, we call the conditions homogeneous, otherwise we call them inhomogeneous.

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

$${\varphi }^{\prime \prime }=0$$

and the solution is just

$$\varphi (x)=ax+b$$

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,

$${\mathrm{\nabla }}^2\varphi =f$$

we start by integrating over the domain

$${\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}^2\varphi d\mathrm{\Omega }={\int }_{\partial \mathrm{\Omega }}\mathrm{\nabla }\varphi \cdot \mathbf{n}dS={\int }_{\mathrm{\Omega }}fd\mathrm{\Omega }$$

If we have homogeneous Neumann BCs on all sides, $\mathrm{\nabla }\varphi \cdot \mathbf{n}=0$, then the source, $f$, must satisfy

$${\int }_{\mathrm{\Omega }}fd\mathrm{\Omega }=0$$

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