Civil Engineering Reference
In-Depth Information
5.9 Lemma.
The spaces X
h
and M
h
satisfy the inf-sup condition (4.16) with a
constant β>
0
if and only if a strengthened Cauchy inequality
1
for all v
h
∈
M
h
,η
h
∈
E
h
(
∇
v
h
,η
h
)
0
,
≤
−
β
2
∇
v
h
0
η
h
0
(
5
.
19
)
holds.
Proof.
Given
v
h
∈
M
h
, by the inf-sup condition there is a
σ
h
∈
X
h
such that
(
∇
v
h
,σ
h
)
0
1. Now for any
η
h
∈
E
h
we conclude from
≥
β
∇
v
h
0
and
σ
h
=
the orthogonality of
X
h
and
E
h
that
∇
v
h
−
η
h
0
≥
(
∇
v
h
−
η
h
,σ
h
)
0
=
(
∇
v
h
,σ
h
)
0
≥
β
∇
v
h
0
.
(
5
.
20
)
Since the strengthened Cauchy inequality is homogeneous in its arguments, it is
sufficient to verify it for the case
η
h
0
=
(
1
−
β
2
)
1
/
2
∇
v
h
0
,
2
0
2
0
2
0
2
(
∇
v
h
,η
h
)
0
=∇
v
h
+
η
h
−∇
v
h
−
η
h
−
β
2
)
∇
v
h
2
0
2
0
−
β
2
)
1
/
2
2
0
2
≤
(
1
+
η
h
=
∇
v
h
η
h
2
(
1
0
,
and the proof of (5.19) is complete.
The converse is easily proved by using the decomposition of
∇
v
h
.
Problems
5.10
Given
σ
∈
H(
div
)
, find
σ
h
∈
σ
−
σ
h
0
is minimal. Charac-
terize
σ
h
as the solution of a saddle point problem (mixed method).
RT
0
such that
5.11
Show that the strengthened Cauchy inequality (5.19) is equivalent to the
ellipticity property
for
v
h
∈
X
h
,η
h
∈
E
h
.
(
∇
v
h
+
η
h
)
2
dx
≥
(
1
2
1
2
−
β)(
|
v
h
|
+
η
h
0
)
Further equivalent properties are presented in Problem V.5.7.
5.12
Define the vectors
a
i
and
b
i
in
2
by
1if
j
=
1 f
j
=
2
i
,
2
i
,
2
−
i
(a
i
)
j
:
=
(b
i
)
j
:
=
if
j
=
2
i
+
1,
0
otherwise,
0
otherwise,
. Show that
A
and
B
are closed, but that
A
+
B
is not. Is there a nontrivial strengthened Cauchy
inequality between the spaces
A
and
B
?
and the subspaces
A
:
=
span
{
a
i
;
i>
0
}
and
B
:
=
span
{
b
i
;
i>
0
}
See also Problem 9.16
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