Biomedical Engineering Reference
In-Depth Information
2.
Definition of test functions
.
q
1
; ; /
on
.0;T/
max
S
: Consider 2
C
c
.Œ0;T/I H
0
.0;1//.Define
8
<
9
=
A constant extension in the vertical
direction of
e
r
on
W
q
1
WD .0; .
z
//
T
I
on
max
n
min
;
;
Notice div
q
1
D 0:
q
1
WD
:
on
min
:
A divergence-free extension to
min
(see, e.g., [
70
,p.127]).
From the construction it is clear that
q
1
is also defined on
t
N
for each N,and
so it can be mapped onto the reference domain by the transformation A
t
N
.
We t a ke 2 H
1
.
S
/ such that .t;
z
;1/D .t;
z
/.
For any test function .
q
; ; / 2
Q
it is easy to see that the velocity component
q
can then be written as
q
D
q
q
1
C
q
1
,where
q
q
1
can be approximated by
divergence-free functions
q
0
that have compact support in
[
in
[
out
[
b
.
Therefore, one can easily see that functions .
q
; /D .
q
0
C
q
1
; /in
X
satisfy the
following properties:
•
X
of all test functions defined on the physical, moving
domain
,definedby(
2.133
); furthermore, r
q
D 0;8
q
2
X
F
.
is dense in the space
Q
•
For each
q
2
X
F
,define
q
D
q
ı A
:
The set f.
q
; ; /j
q
2
X
F
; 2
X
S
; 2
X
S
g is dense in the
D
q
ı A
;
q
Q
of all test functions defined on the fixed, reference domain
F
,defined
by (
2.138
).
space
For each
q
2
X
F
,define
•
q
N
WD
q
ı A
t
N
:
t
N
Functions
q
N
are defined on the fixed domain
F
, and they satisfy r
q
N
D
0.
Functions
q
N
will serve as test functions for approximate problems associated
with the sequence of domains
t
N
, while functions
q
will serve as test functions
associated with the domain
. Both sets of test functions are defined on
F
.
Lemma 2.6.
For
every
.
q
; ; /
2
X
we
have
q
N
!
q
uniformly
in
L
1
.0;T I C.
F
//
.
Proof.
By the Mean-Value Theorem we get:
j
q
N
.t;
z
;r/
q
.t;
z
;r/jDj
q
.t;
z
;.1C
t
N
/r/
q
.t;
z
;.1C /r/j
Dj@
r
q
.t;
z
;/rjj.t;
z
/
N
.t t;
z
/j:
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