sci.math.num-analysis,sci.philosophy.tech

SUNA01, The Manifesto
=====================

The area of numerical mathematics is splitted up in two distinct disciplines,
called respectively the Finite Difference method (F.D.) and the Finite Element
method (F.E.). This fact is denied, by declaring that finite differences are
merely a special case of finite elements (: O.C. Zienkiewicz), or trivialized,
by "ignoring the siren voices from the finite element champ" (: D.B. Spalding).
The contradiction is too profound to be neglected, however.

Systems of coupled partial differential equations of considerable complexity,
such as those describing fluid flow and heat transfer, can be made accessible
to numerical treatment. Finite Difference Methods, but even more specificially
Finite Volume Methods such as those described in [1], are very successful in
this area. The robustness of the F.V. discretization schemes, employing just
"Four Basic Rules", has no counterpart in Finite Element methodology.
The main drawback of Finite Difference -like methods is well known, however.
When attempting to get rid of inhomogeneous parts of the calculation domain,
caused for example by curved boundaries, a considerable overhead is introduced,
tending to make F.D./F.V. methods unworkable.

When employing a Finite Element Method, curvilinear boundaries and topological
complexity present no problem whatsoever. They are done in a uniform and natural
fashion, which has no counterpart in Finite Difference methodology. This at last
partly explains why F.E. methods have become so widely used in solid mechanics,
where an accurate description of geometry and connectivity is important [2].
The main drawback of Finite Element Methods is well known, however. In order to
formulate an F.E. discretization scheme properly, something like a variational
or Galerkin principle has to be resorted to. When dealing with very complicated
equations, especially those describing transport phenomena, this turns out to
be a serious bottleneck.

It is observed that the weak points of Finite Differences could be covered very
precisely by the advantages of a Finite Element technique. The reverse is also
true: certain drawbacks of a Finite Element method could be compensated easily,
if it only were possible to use a Finite Difference approach in its context.
All this cannot be true by coincidence. Let it be stated here very firmly that
the existence of two (and more) separate numerical methods cannot be justified,
neither from a scientific, nor from a practical point of view. If it were not
possible to understand the nowadays situation within its historical context,
then it could not be understood at all.

Unification principle
---------------------
| There should be ONE numerical method, instead of two (and more).
| This method should be such that say Finite Differences and Finite Elements
| are nothing else than just two different ways of looking at the same thing.
| In other words, F.D. and F.E. are the two dual aspects of one and only one
| universal numerical method.

This Unification principle, instead of being merely a matter of philosophical
consideration, will be shown in the sequel to imply far reaching mathematical
consequences. For the moment being, it has to be considered as a basic axiom.
There should be no doubt about the fact that Unified Numerical Approximations
can be constructed, wishfully, on purpose, as an act of free will. The future
is not what will happen to us, but: what we shall do.

SUNA References:
1. S.V. Patankar; Numerical Heat Transfer and Fluid Flow; Hemisphere Publishing
   Company U.S.A. 1980.
2. O.C. Zienkiewicz; The Finite Element Method; 3th edition; Mc.Graw-Hill U.K.
   1977.

To be continued in 'sci.math.num-analysis' and 'sci.math.research' ...
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