Biomedical Engineering Reference
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Crp; ..v
z
n
C
1
/ r/' Crq
;
K
ı
K
v
n
1
/;..v
z
n
C
1
/ r/' Crq
F
h
.V/ D
X
K
2
T
h
2t
.4v
n
;
(5.23)
K
U D .v;p/; V D .';q/; U
D .v
;p
/;
where ı
K
0 are suitable parameters. Moreover, the additional div-div stabilization
form
P
h
.U;V/ D
X
K
2
T
h
K
.rv; r'/
K
(5.24)
is introduced with suitable parameters
K
0.
The stabilized discrete problem reads: Find U
h
D .v
h
;p
h
/ 2 W
h
Q
h
such that
v
h
satisfies approximately conditions (
5.4
), (a), (b), (c)(i) and
a.U
h
;U
h
;V
h
/ C
L
h
.U
h
;U
h
;V
h
/ C
P
h
.U
h
;V
h
/ D f.V
h
/ C
F
h
.V
h
/
for all V
h
D .'
h
;q
h
/ 2 X
h
Q
h
:
(5.25)
Stabilization Parameters
The choice of the parameters ı
K
and
K
is carried out according to [
53
]and[
82
].
The parameter ı
K
is defined on the basis of the local transport velocity v
z
n
C
1
and
local element size h
K
of K measured in the direction of the vector .v
z
n
C
1
/.b
K
/,
where b
K
denotes the barycenter of K. In the case of the Taylor-Hood finite
elements the following choice of parameters appears suitable:
ı
K
D ı
h
2
K
;
K
D
;
(5.26)
where
>0and ı
>0are fixed constants.
The fully stabilized problem allows also the application of the equal-order P
1
=P
1
finite elements, which do not satisfy the BB condition. In this case, we set D =
and introduce the parameters
K
D
1 C Re
loc
;
h
2
K
t
h
2
K
K
C
ı
K
D
;
(5.27)
where the local Reynolds number Re
loc
is defined as
h
K
k
v
k
K
2
Re
loc
D
:
(5.28)
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