the boundary corrector

Start with the free-space response to a point source — the fundamental solution. It solves the equation correctly inside, but it knows nothing about the walls of your region, so on the boundary it has the wrong value. The boundary corrector is the patch: a smooth, source-free field you add on top so that the total exactly meets the boundary condition, without disturbing the point source inside.

Write the Green's function as G(x, y) = E(x - y) + h(x, y). Here E is the fundamental solution, which already gives L E = delta (the point source) and carries all the singularity. The corrector h(x, y) is required to be a homogeneous solution inside, L h = 0 (for the Laplacian, h is harmonic), so that adding it does not spoil L G = delta. Its job is purely on the boundary: it is chosen so that on the wall, h(x, y) = -E(x - y) for a Dirichlet condition (making G = 0), or so the normal derivatives cancel appropriately for a Neumann condition. So the corrector is itself the solution of a boundary-value problem — a homogeneous equation with boundary data equal to minus the fundamental solution's boundary trace. Finding h is the entire difficulty of building a Green's function.

Two things are worth stressing. The corrector is smooth where the fundamental solution is singular: all the blow-up lives in E at the source point, and h is well-behaved there, which is why h is also called the regular part of the Green's function. And the method of images is nothing but a slick way to write h: instead of solving a boundary-value problem for the corrector, you place fictitious image sources outside the region whose combined free-space field happens to equal h inside — but that trick only works in geometries symmetric enough to admit images.