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Re: The relaxation method



On Fri, 1 Mar 2002, Ludwik Kowalski wrote:

I see that John's equipotentials are everywhere perpendicular
to the paper borders. How did you relax marginal cells (those
which have only three or two neighbors), John, to get this?

Tempted as I may be to answer this question, John Denker has
already proposed an alternate approach based on a model of good
pedagogy that I am loathe to undermine. Perhaps I will just
provide the following hint which will allow you an alternate way
of thinking about JD's "periodic boundary conditions".

Laplace's equation is very well known to have unique solutions as
long as appropriate boundary conditions are specified.
Appropriate boundary conitions come in two forms:

1. "Dirichlet": Setting the value of the potential itself on the
boundary.

2. "Neumann": Setting the normal derivative of the potential at
the boundary.

You are familiar with the Dirichlet case but are struggling with
the Neumann case which is needed here. So here's the hint: How
would you specify the value of a marginal cell to insure that the
normal derivative of the potential is zero at the boundary?

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm