sci.math.numanalysis
SUNA32, The 1983 Colloquium at CWI
==================================
Reference is made to a reprint from the "Colloquium Topics in Applied Numerical
Analysis" by J.G. Verwer, at the Centre for Mathematics and Computer Science,
Amsterdam, the Netherlands (1984). It was Christmas time. My own contribution
is called "Least squares numerical analysis of the steady state and transient
thermal hydraulic behaviour of L.M.F.B.R. heat exchangers". LMFBR= Liquid Metal
Fast Breeder Reactor, the thing they were building at that time in Kalkar, West
Germany. The Duth partner in this project was Neratoom, my previous employer.
Use will be made of the theorem which was proved in SUNA31. It will be applied
however on systems of equations S which obey a stronger set of rules, namely:
(i) For all i : S(i,i) > 0 ; S(i,j) <= 0 for all j <> i .
(ii) sum S(i,j) >= 0 , with strict inequality for some i .
i
(iii) S is irreducible.
It can easily be shown that the above conditions are stronger than those needed
to prove the theorem in SUNA31:
From (i) it follows that: S(i,i) <> 0 .
From (ii) it follows that: S(i,i) >= sum  S(i,j)
j<>i
With the help of (i):  S(i,i)  >= sum  S(i,j) 
j<>i
I want to talk about a significant paragraph in the abovementioned paper called
"Stability". It starts on page 182 of the proceedings. Our theory is worked out
in the paper for a more general case, but attention will be restricted in this
article to my Pure Convection hobby horse. Formulated in local coordinates as:
U.dT/dh + V.dT/dk = 0
The analytical approximation (yes) is discretized upon the parent quadrilateral
: a square. It's shape functions are: (1h).(1k) , h.(1k) , (1h).k , h.k
1 2 3 4
/ Differentiation:
3 o  o 4
 /  d/dh = [ (1k) +(1k) k +k ]
 * 
 /  d/dk = [ (1h) h +(1h) +h ]
1 o X o 2
U.dT/dh + V.dT/dk = [  U.(1k)  V.(1h) ].T1
+ [ + U.(1k)  V.h ].T2
+ [  U.k + V.(1h) ].T3
+ [ + U.k + V.h ].T4 = 0
Here (h,k) are the coordinates of integration point(s), the position of which
will be established in the sequel.
Let us assume for simplicity that: U > 0 , V > 0 , U < V . This situation will
be depicted graphically as in the above ASCII figure.
A more serious restriction can be imposed by demanding that, in case of U > 0
or V > 0 , the twodimensional description automatically simplifies to a one
dimensional theory. This is accomplished most easily by positioning integration
points along the streamline or the characteristic going through a nodal point.
In such a way, however, that the coordinates (h,k) lie _inside_ the element.
In the situation sketched above, only the streamlines through (1) and (4) are
to be selected. But of course only _upwind_ temperatures can be felt by each of
these nodes, which already obsoletes node (1). Hence, any integration point can
only be placed at the streamline through (4). To be more precise, coordinates
(h,k) must obey the relationship:
(k1) = V/U.(h1) or: U.(1k) = V.(1h)
To proceed further, let us bring in mind Ames' theorem, as has been formulated
in this article. The discretization is considered as an equation for node (4).
It can be checked out easily that condition (ii) is automatically fulfilled.
But according to (i) the following inequalities must hold:
 U.(1k)  V.(1h) <= 0 : T1
+ U.(1k)  V.h <= 0 : T2
 U.k + V.(1h) <= 0 : T3
+ U.k + V.h > 0 : T4
Because of 0 <= h,k <= 1 ; U,V > 0 , the last condition is fulfilled without
question. Looking at the figure, it is seen that the nodal characteristic is
embedded in a domain of influence, which is thought to be bounded by the lines
(1)  (4) and (2)  (4). Therefore it is suggested that, physically speaking,
(4) obtains temperature information from (1) and (2) but will not be influenced
by (3), which is outside the skewed domain. For this reason, it would certainly
be a good idea to set the corresponding matrix coefficient to zero:
 U.k + V.(1h) = 0 : T3
U.(1k) = V.(1h) : because (*) on streamline
Solution: k = 1k > k = 1/2 ; 1h = U/V/2 > h = 1  U/V/2
Substitution into the parent quadrilateral upwind scheme gives:
[  U/2  U/2 ].T1
+ [ + U/2  V + U/2 ].T2
+ [ 0 ].T3
+ [ + U/2 + V  U/2 ].T4 = 0
Which is very much the same as: U.(T2  T1) + V.(T4  T2) = 0
But the latter equation is recognized as a specialization of 2D SUFED: SUNA05
contains an equation (6) which resembles the above one.
Conclusions:

While devising an upwind parent quadrilateral it is found that the accompanying
discretization scheme is completely equivalent with Skew Upwind Finite Element
Differences at a triangle. Significant detail is that an F.E. integration point
must lie in the middle of the piece of the streamline that crosses the element:
^^^^^^
o
 \
 \
*o * = integration point
 /
 /
o
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