Biomedical Engineering Reference
In-Depth Information
Then we define the approximate solution as a couple 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 the identity
a.U
h
;U
h
;V
h
/ D f.V
h
/; for all V
h
D .'
h
;q
h
/ 2 X
h
Q
h
:
(5.20)
The couple .X
h
;Q
h
/ of the finite element spaces has to satisfy the Babuška-
Brezzi (BB) inf-sup condition (see, e.g., [
35
,
36
]or[
99
]), i.e., we assume that there
exists a constant LJ>0such that
.p; rv/
krvk
L
2
./
LJkpk
L
2
./
;
sup
v
2
X
h
8p 2 Q
h
:
(5.21)
In practical computations we assume that the domain is a polygonal approx-
imation of the region occupied by the fluid at time t
n
C
1
. The spaces W
h
;X
h
;Q
h
are defined over a triangulation
T
h
of the domain , formed by a finite number of
closed triangles K 2
T
h
with the following properties:
(a)
D
S
K
2
T
h
K,
(b)
the intersection of two different elements K;K
0
2
T
h
is either empty or a
common edge or a common vertex of these elements,
(c)
the vertices lying on @ belong to @
t
n
C
1
,
(d)
the end points of
I
;
O
,and
Wt
are vertices of some elements K 2
T
h
.
We shall denote by h
K
the length of the maximal side and we assume the index h
is chosen as h D max
K
2
T
h
h
K
. The spaces W
h
;X
h
,andQ
h
are formed by piecewise
polynomial functions. In our computations, the well-known Taylor-Hood P
2
=P
1
conforming finite elements are used for the velocity/pressure approximation. This
means that p
h
is a linear function and v
h
is a quadratic vector-valued function on
each element K 2
T
h
, i.e., the spaces W
h
, X
h
,andQ
h
are defined by
H
h
Df' 2 C./I 'j
K
2 P
2
.K/ for each K 2
T
h
g;
W
h
D ŒH
h
2
;
X
h
D W
h
\ X;
(5.22)
Q
h
D
ǚ
' 2 C./I 'j
K
2 P
1
.K/ for each K 2
T
h
;
where P
k
.K/ denotes the space of all polynomials on K of degree less or equal to
k.
The standard Galerkin discretization (
5.20
) may produce approximate solutions
suffering from spurious oscillations for high Reynolds numbers. In order to avoid
this drawback, the stabilization via streamline-diffusion/Petrov-Galerkintechnique
is applied (see, e.g., [
34
,
53
]). The stabilization terms are defined as
ı
K
3
L
h
.U
;U;V/D
X
K
2
T
h
T
v/ C
.v
z
n
C
1
/ r
v
2t
v r.rv Cr
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