Civil Engineering Reference
In-Depth Information
Nearly Incompressible Flows
Instead of directly enforcing that the flow be divergence-free, sometimes a penalty
term is added to the variational functional
1
2
[
(
∇
v)
2
+
t
−
2
(
div
v)
2
−
2
fv
]
dx
−→
min!
Here
t
is a parameter. The smaller is
t
, the more weight is placed on the restriction.
In this way a nearly incompressible flow is modeled.
The solution is characterized by the equation
a(u, v)
+
t
−
2
(
div
u,
div
v)
0
,
=
(f, v)
0
,
for all
v
∈
H
0
()
n
.
(
6
.
13
)
In order to establish a connection with the standard formulation (6.5), we set
p
=
t
−
2
div
u.
(
6
.
14
)
Now (6.13) together with the weak formulation of (6.14) leads to
for all
v
∈
H
0
()
n
,
a(u, v)
+
(
div
v, p)
0
,
=
(f, v)
0
,
(
6
.
15
)
(
div
u, q)
0
,
−
t
2
(p, q)
0
,
=
for all
q
∈
L
2
,
0
()
n
.
0
Clearly, in comparison with (6.5), (6.15) contains a term which can be interpreted
as a penalty term in the sense of §4. By the theory in §4, we know that the solution
converges to the solution of the Stokes problem as
t
→
0.
Problems
6.6
Show that among all representers of
q
∈
L
2
()/
R
, the one with the smallest
inf
c
∈
R
q
+
c
0
,
is characterized by
qdx
=
L
2
-norm
q
0
,
=
0. [Conse-
quently,
L
2
()/
R
and
L
2
,
0
()
are isometric.]
6.7
Find a Stokes problem with a suitable right-hand side to show that for every
q
∈
L
2
,
0
()
, there exists
u
∈
H
0
()
with
div
u
=
q
and
u
1
≤
c
q
0
,
where as usual,
c
is a constant independent of
q
.
6.8
If
is convex or sufficiently smooth, then one has for the Stokes problem
the regularity result
u
2
+
p
1
≤
c
f
0
;
(
6
.
16
)
see Girault and Raviart [1986]. Show by a duality argument the
L
2
error estimate
u
−
u
h
0
≤
ch(
u
−
u
h
1
+
p
−
p
h
0
).
(
6
.
17
)
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