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Transcript
```Multiplying (A.1) and (A.2) through by test functions and integrating by parts yields
(h, a)
/
- (#h,V - a)T +
(Trh, a),T
h, a) T
-(V
KEh
T
n,"-),Th
-
h,
-
a -n
OT
= 0,
Th, T
(Tq~h, P) a~h +
)~h=
(A.4)
(-pco, a)T,,
(A.5)
(g, w' )Th~
(A.6)
Note that the numerical flux given by equation (A.3) has been substituted into the
system. This is possible since the problem is linear. The resulting elemental matrix
system has the form
AK
-BK
BT
DK
CK
EK
--
CK
QK
-EK
UK
MK
0
(A7)
=FK
JK
GK
The upper left quadrant is again block diagonal and may be inverted to reduce the
system to degrees of freedom on the boundaries. Note that the system in this case
has fewer unique sub-matrices. This facilitates matrix assembly by reducing required
computations. The final form of the global problem is
HO = R.
(A. 8)
Newton iterations are not required since the problem is linear. The accuracy of the
solution is then limited by the mesh and element order.
80
```
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