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SUNA13, Theorem & Corollaries
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This article is part of the "Series on Unified Numerical Approximations"
UNIFICATION OF FINITE VOLUME AND GALERKIN METHODS
FOR CONSERVATION EQUATIONS, IN THE TWO DIMENSIONAL CASE
PART III : THE THEOREM
In the previous article, an almost completed proof was given for the following
First Unification Theorem

 There exists an EQUIVALENCE between Finite Domain methods at Five Point Stars
 and Galerkin methods at Quadrilaterals, when both applied to the conservation
 equation in two dimensions, as formulated in general curvilinear coordinates.

 A necessary condition for the Galerkin method is, however, that the latter is
 carried out with numerical integration at the _vertices_ of its Quadrilateral
 elements (with is rather unusual).

 This unusual integration procedure is equivalent, in turn, with discretizing
 at a set of mutually overlapping linear triangular elements. The integration
 points of them must lie at the vertices of corresponding quadrilaterals.
Suppose namely that we decide to evaluate the localized Galerkin integral at a
linear triangle, which must be an even simpler procedure, then:
dF/dh = F2  F1 ; dF/dk = F3  F1 ; and the integral equals:
  Qh  Qk 
  F1 F2 F3  .  + Qh 
 Qk 
Here the Q's are evaluated at the (only) integration point of the triangle.
Part of the corresponding formula in "SUNA, Theorem Proving" reads:
h = 0 , k = 0 3 *  * 4 Triangles:
  \ /  1,2,3
  Qh1  Qk1   \ /  2,4,1
 1/4. [  + Qh1  0  + ...  / \  3,1,4
  0 + Qk1   / \  4,3,2
 + 0 + 0  1 *  * 2
The two terms are equal iff the integration point of the triangle attached at
the quadrilateral node (1) precisely coincides with node (1); same argument for
all of the four triangles.
It is concluded therefrom that a quadrilateral element, with integration points
at its corners, can entirely be replaced by a set of four overlapping triangles
with their integration points at the vertices of the quadrilateral.
The F.D.F.E. Unification of the 2D conservation equation problem as a _whole_
thus requires two, mutually _overlapping_, meshes of triangles, equipped with
rather unusual integration points.
This completes the proof of the First Unification Theorem.
Some consequences:
 The F.E. problem how to make the "best" choice between different triangular
patterns in a rectangular grid is circumvented by the F.D. point of view:
there exists *no* preferred pattern within a mesh of overlapping triangles.
 Finite Differences are intrinsically more *in*efficient than Finite Elements,
since they require a subdivision of quadrilaterals into *four* triangles,
while F.E. normally does its job with only *two* triangles per quadrilateral.
(This will become noticable only in the curvilinear case, however, since the
way in which the computations are actually organized plays a dominant role.)
 The fact that Finite Element meshes can be built up from overlapping elements
suggests that even more unusual topologies *can* result in numerical schemes
that still work, and there is a chance that they sometimes are even better.
For pure Diffusion of heat, approximations of the fluxes (Qh,Qk) are constants.
Then the position of the triangle integration points becomes _irrelevant_, and
only the idea of overlapping elements survives.
To be continued ...

* Han de Bruijn; Applications&Graphics  "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354  would be NO idleness in * Oil
* 2600 AJ Delft; The Netherlands.  Mathematics" (HdB). * for
* Email: Han.deBruijn@RC.TUDelft.NL  Fax: +31 15 78 37 87 * Blood