I had a different objective in mind when I took up the task of
calculating a potential field on a spreadsheet. I wanted a
pedagogically useful algorithm which exemplifies the geometrical
meaning of Laplace's equation in the most transparent way
possible. I did not intend to discover the most rapidly
convergent algorithm for calculating the field.
The geometrical meaning of Laplace's equation (div grad V = 0) is
that in free space the potential V at any point is equal to the
average potential on the surface of any sphere centered on that
point. When we partition space into cells we can take crude
averages in the manner described earlier. The problem is simple
in the case of Cartesian coordinates, but symmetrical problems
can be approached more easily by exploiting the symmetry. (No one
has yet suggested the one-dimensional case, which corresponds to
spherical symmetry, but that, too is worthy of consideration.) I
have been trying to construct my algorithm with the pedagogical
goal in mind. In doing a practical calculation in which the sole
object is to find the answer, of course, other methods are to be
preferred.