# The Manifesto

The area of Numerical Analysis is splitted up in two distinct disciplines, called the Finite Difference method (F.D.) and the Finite Element method (F.E.) respectively. This fact is denied, by declaring that Finite Differences are merely a special case of Finite Elements (: O.C. Zienkiewicz in [ 1 ] ), or trivialized, by "ignoring the siren voices from the finite element champ" (: D.B. Spalding in [ 2 ] ). But, after all these years, the war still has to come to an end.

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 [ 3 ], 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 [ 1 ] . 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 a F.E. 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 and social context, then it could not be understood at all.

## Unification principle

There should exist one science, called Numerical Analysis, instead of two (and more) separate disciplines. This science should be such that say Finite Differences and Finite Elements are nothing else than just two different ways of looking at the same thing. With other words, F.D. and F.E. are the two dual aspects of one and only one universal piece of mathematics.

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.

## A fairy-tale

Numerical Analysis is the Chinderella of Mathematics. She is good enough for the hard work to be accomplished. But it seems that (almost) nobody is aware of her very special beauty. She always has to obey her stepsisters (among them are Banach and Sobolev). Her Upwind Schemes are judged to be inaccurate; some of her best elements are called non-conformal. By "deriving" ridiculous upper bounds for her accuracy, her stepmother (Classical Analysis) seems to have established supremacy for indefinite time. But everybody knows the outcome of this fairy-tale: the elegant slippers of glass will only fit to the feet of Chinderella.

## References:

1. O.C. Zienkiewicz; The Finite Element Method; 3th edition; Mc.Graw-Hill U.K. 1977.
2. D. Brian Spalding; A General Purpose Computer Program for Multi-Dimensional One- and Two-Phase Flow; Mathematics and Computers in Simulation, North Holland Press, vol XXIII, pp 267-276, 1981.
3. S.V. Patankar; Numerical Heat Transfer and Fluid Flow; Hemisphere Publishing Company U.S.A. 1980.