From: fme@aaargo.ttc-cmc.net (Fred Erickson)
Subject: Tutorial on numerical analysis of potential and capacitance
Date: 24 Dec 2000 23:12:58 -0700
Newsgroups: sci.physics,sci.math.num-analysis
Summary: [missing]

As of December 24, 2000, an updated version of my 56-page tutorial
on the numerical analysis of potential and capacitance is available
(in postscript format) at:

http://www.ttc-cmc.net/~fme/captance.html

This is an introduction for beginners, and probably won't be of any
value to experienced analysts.  The paper is provided free of charge,
and is not intended as an advertisement for any product or service.
It is based entirely on conventional physics, to the extent that I
understand conventional physics, and is not intended to promote some
new theory.  I worked long and hard on it, mostly to clarify my own
understanding, and I offer it to the world at large in the hope
that it may prove useful.  The new version isn't radically different
from  the old one, but does include some new material to provide a
better understanding of Jacobi iteration, and fixes a few minor things
that I wasn't happy with.

Using the non-ideal parallel-plate capacitor as a central theme, the
document introduces Conformal Mapping, Finite Difference Methods,
Finite Element Methods, and Boundary Element Methods.  Readers are
assumed to have some familiarity with calculus and differential
equations, but shouldn't need a lot of specialized background knowledge
to understand most of the document.  (It would, of course, be useful to
have some prior knowledge of electrostatics, but the basic equations
of electrostatics are reviewed in the introduction.)  In the discussion
of the Finite Element Method, some use is made of concepts from vector
calculus.  People who have never multiplied or divided complex numbers,
or who are not comfortable with the concept of a function of a complex
variable, may want to skip the Conformal Mapping section.

Be warned.  I am not an expert on numerical analysis, or on mathematics
in general.  Some time ago, I decided to use the parallel-plate capacitor
problem as a way to learn some numerical techniques, purely for my own
enjoyment.  This document is my attempt to desribe what I learned, on the
theory that, if I can't explain it to someone else, then I don't understand
it myself.  I am basically an advanced beginner, trying to explain things
to someone who is just getting started.  Thus, I may not use the most
up-to-date terminology, and I certainly don't present the most advanced
algorithms available, nor even the most advanced algorithms of which I am
aware.  Despite my best efforts, the document may contain errors.

Note also, that mathematical terminology can be slippery.  As near as I can
tell, Boundary Element Methods may also be called by names like:  the
Method of Moments, Boundary Integral Methods, Boundary Integral Equation
Methods, Spectral Methods, Pseudo-Spectral Methods, Integral Equation
Methods, and possibly also by other names (some perhaps profane) that I
haven't come across.  The terminology depends on the details of the
specific approach, on the backgrounds of the people involved, and on
the assumed level of abstraction.  Thus, this document should not be
considered a definitive source when it comes to terminology.  (I doubt
that such a source exists.)

The document confines itself purely to electrostatics, and assumes
that no dielectrics are present.  However, an understanding
of the electrostatic case, using Laplace's equation, should help
prepare you for other cases and other equations.  If nothing else,
you may learn something about capacitors.  I know I did.

The web site also contains a second paper, which derives the capacitance
between two unequal-sized spheres, and explores some ambiguities in the
definition of the word ``capacitance''.  This second paper should be
comprehensible to people with a basic background in analytic geometry.

Fred Erickson,  email address available at the above web site