For a fixed differential equation, generically in 3D space, are there any theorems, or lore, on whether, the number of points at which the system is being computed affects either the required word length (i.e. precision), or number of iterations needed to get to an accurate answer?
a priori it would seem that the precision needed will only depend on how many neighbor points are coupled to a given point, and I would expect that to be the strongest effect. But too one might hazard that as the number of points gets large, the precision would need to go up as well. Iterations seems more straight forward: as the number of points goes up, its going to take more iterations to settle down into the true behavior, but that's just my intuition, I'm wondering if anyone has any experience and actual data with this kind of question...
Thanks in advance.
You are asking interesting and relevant questions. There are certain calculations you should do and certain texts that you should read.
In particular, you should compute the condition number for the discrete Laplacian in a single dimension. You will find that you need a small unit round off error when the number of point is large. It will not be hard to extend this computation to higher dimensions.
Next you should study the convergence rate of some of the classical iterative methods: Jacobi, Gauss-Seidel, and successive overrelaxation. Since you are interested in differential equations, I recommend
Arieh Iserles: "A first course in the numerical analysis of differential equations".
In particular, this book has a nice section on iterative solution of Poisson's equation. You will see that a larger number of iterations is indeed required when you increase the number of points.
However, this only quantifies the effects. It does not really explain why we need a smaller round off error or a large number of iterations, when we increase the number of points.
Compare this with the problem of stating Newton's law of gravity versus the problem of explaining the nature of gravity.
Your real questions are much harder to explain and more advanced mathematics is (probably) required. In particular, knowlegde of unbounded linear operators on infinite dimensional Hilbert spaces is certainly useful. Here I can recommend
E. B. Davies: "Spectral theory and differential operators".
In particular, the first chapter has a nice discussion of the one dimensional Laplace operator.
In a nutshell: As you increase the number of points your discrete operator becomes a better approximation of a "nasty" operator and it is no surprise that the "nasty" properties are passed on to the approximation, yielding a matrix which is very ill conditioned.
This will put you on the path to understand why a smaller unit round off error is required.
As for the second half of your real question, i.e., why the iteration count goes up, I do not have good explanation at this time.