Suppose we want to solve the Laplacian $\nabla^2\phi=0$ with $\phi$ vanishing at infinity. Suppose also that we want an extra boundary condition, say $\phi=1$ on some bounded region $R$ of space.
The PDE solving software package I'm using (FiPy) doesn't support this. It only supports solving the Laplacian with boundary conditions in a region of genus 0. However, they recommend getting around this by "adding source terms". The linked documentation is very unclear to me, but if I understand correctly, they're claiming that the original problem is approximately equivalent to solving:
$$\nabla^2\phi(x) - K1_R(x)\phi(x) + K1_R(x)=0\tag 1$$
Where $1_R$ is the indicator of $R$ and $K$ is some very large positive value, and $\phi$ vanishes at infinity. Why is this approximately equivalent to solving the original problem?
Note: Like I said, I'm not sure at all that (1) really is the equation they're getting at, but I think it's something close to that. I'm hoping someone can correct me if I'm misinterpreting the documentation.