Civil Engineering Reference
In-Depth Information
4.12 Example.
Let
X
=
M
:
=
L
2
()
,
a(u, v)
:
=
0,
b(v, µ)
:
=
(v, µ)
0
,
, and
c(λ, µ)
:
=
K
·
(λ, µ)
0
,
. Clearly, the solution of
b(v, λ)
=
(f, v)
0
,
,
b(u, µ)
−
t
2
c(λ, µ)
=
(g, µ)
0
,
is
λ
=
f
and
u
=
g
+
t
2
Kf
. Thus, the solution grows as
K
→∞
and we cannot
expect a bounded solution for an unbounded bilinear form
c
.
In plate theory we frequently encounter saddle point problems with penalty
terms which represent
singular perturbations
, i.e., which stem from a differential
operator of higher order. Then we introduce a semi-norm on
M
c
, and define the
corresponding norm
c(µ, µ),
|
µ
|
c
:
=
(
4
.
22
)
|||
(v, µ)
|||
:
=
v
X
+
µ
M
+
t
|
µ
|
c
,
on
X
×
M
c
; see Huang [1990]. On the other hand, this now requires the ellipticity
of
a
on the entire space
X
, rather than just on the kernel
V
as in Theorem 4.3. It is
clear from the previous example that we indeed need some additional assumption
of this kind.
4.13 Theorem.
Suppose the hypotheses of Theorem 4.3 are satisfied and that a is
elliptic on X. Then the mapping L defined by the saddle point problem with penalty
term satisfies the inf-sup condition
A(u, λ
;
v, µ)
|||
(u, λ)
||| · |||
(v, µ)
|||
≥
γ>
0
,
inf
(u,λ)
sup
(v,µ)
∈
X
×
M
c
(
4
.
23
)
∈
X
×
M
c
for all
0
1
, where γ is independent of t.
These two theorems are consequences of the following lemma [Kirmse 1990]
whose hypotheses appear to be very technical at first glance. However, by Prob-
lem 4.23, the condition (4.25) below is equivalent to the Babuska condition for
the
X
-components,
≤
t
≤
A(u,
0
;
v, µ)
|||
(v, µ)
|||
≥
α
u
X
,
sup
(v,µ)
(
4
.
24
)
with suitable
α
. In particular, it is therefore also necessary for stability.
4.14 Lemma.
Suppose that the hypotheses of Theorem 4.3 are satisfied, and sup-
pose that
a(u, u)
u
X
+
b(u, µ)
µ
M
+
t
|
µ
|
c
≥
α
u
X
sup
µ
∈
M
c
(
4
.
25
)
for all u
∈
X and some α>
0
. Then the inf-sup condition (4.23) holds.
Search WWH ::
Custom Search