Biomedical Engineering Reference
In-Depth Information
Fig. 2.13
Asketchofthe
fluid domains in STEP 1
Z
T
Z
max
j
u
N
N
u
N
2
j
0
D .
Z
T
0
Z
Z
T
Z
/j
u
N
N
u
N
2
:
.t/
4
t
N
.t/
C
j
0
.t/
\
t
N
.t/
Here AB WD .A [ B/ n .A \ B/.SeeFig.
2.13
. Because of the uniform
convergence (
2.169
) we can make the measure j
.t/4
t
N
.t/j arbitrary small.
Furthermore, by Propositions
2.5
and
2.6
we have that the sequence .
u
N
N
u
N
/
N
2N
is uniformly bounded in L
2
..0;T/
max
/: Therefore, for every ">0,
there exists an N
0
2
N
such that for every N N
0
we have
Z
T
Z
2
<
"
.t/
4
t
N
.t/
j
u
N
N
u
N
j
2
:
(2.171)
0
To estimate the second term, we need to measure the relative difference between
the function
u
N
composed with A
1
.t/, denoted by
u
N
, and the same function
u
N
composed with A
1
t
N
.t/
, denoted by
u
N
. We will map them both on the same
domain and work with one function
u
N
, while the convergence of the L
2
-integral
will be obtained by estimating the difference in the ALE mappings. More precisely,
we introduce the set ! D A
1
.
.t/ \
t
N
.t//
F
.Now,weusethe
properties of the ALE mapping A
and the definitions of
u
N
;
u
N
to get
Z
T
Z
Z
T
Z
1
1 C
j
u
N
u
N
ı A
1
.t/
\
t
N
.t/
j
u
N
N
u
N
2
2
j
D
t
N
.t/
ı A
.t/
j
0
0
!
Z
T
Z
1
1 C .t;
z
/
j
u
N
.t;
z
;r/
u
N
.t;
z
;
1 C .t;
z
/
1 C
t
N
.t;
z
/
r/j
2
D
0
!
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
Z
T
Z
@
r
u
N
.t;
z
;/r
1
2
1 C .t;
z
/
1 C
t
N
.t;
z
/
D
0
!
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