🚀 go-pugleaf

LaPlace's equation can be solved on a rectangular region using a 5 point
difference formula or a 9 point difference formula at every point. 

However if there is an object with curved boundaries inside the rectangular
region (dirichlet boundary conditions) the difference formula has to be
modified at points adjacent to the boundaries. 

There are two approaches:
1. use a modified difference equation where the 'template' has unequal
distances up/down/left/right so that some of the 5 points are placed
on the actual curved boundary. I have this formula.
2. use a linear extrapolation between the actual boundary and points not next
to the boundary. This extrapolation can be along the x-direction, y-direction
or a combination of both. I can do this. It works as long as the extrapolated
points are not subject to over-relaxation. In fact for my problem, a set of
mulipoles, it appears to work better than 1. above, which is recommended as
being more accurate.

The question is: are there equivalent formulas for the 9-point case. I have been
looking for these for some time without much luck.

P.S. Another approach I think might be promising is harmonic extrapolation.
Thanks in advance Richard Rothwell