Civil Engineering Reference
In-Depth Information
Saddle Point Problems with Penalty Term
To conclude this section, we consider a variant of Problem (S) which plays a role
in elasticity theory. We want to treat so-called
problems with a small parameter t
in such a way that we get convergence as
h
→
0 which is uniform in the parameter
t
. This can often be achieved by formulating a saddle point problem with penalty
term. Readers who are primarily interested in the Stokes problem may want to
skip the rest of this section.
Suppose that in addition to the bilinear forms
a
and
b
,
c
:
M
c
×
M
c
−→ R
,
c(µ, µ)
≥
0
for all
µ
∈
M
c
(
4
.
20
)
is a bilinear form on a dense set
M
c
⊂
M
. Moreover, let
t
be a small real-valued
parameter. Now we modify (4.4) by adding a
penalty term
:
Problem (S
t
).
Find
(u, λ)
∈
X
×
M
c
with
a(u, v)
+
b(v, λ)
=
f, v
for all
v
∈
X,
(
4
.
21
)
b(u, µ)
−
t
2
c(λ, µ)
=
g, µ
for all
µ
∈
M
c
.
The associated bilinear form on the product space is
=
a(u, v)
+
b(v, λ)
+
b(u, µ)
−
t
2
c(λ, µ).
A(u, λ
;
v, µ)
:
First we consider the case where
c
is bounded [Braess and Bl omer 1990].
Then
c
can be extended continuously to the entire space
M
×
M
, and we can
suppose
M
c
=
M
.
4.11 Theorem.
Suppose that the hypotheses of Theorem 4.3 are satisfied and that
a(v, v)
≥
0
for all v
∈
X. In addition, let c
:
M
×
M
−→ R
be a continuous
bilinear form with c(µ, µ)
≥
0
for all µ
∈
M. Then (4.21) defines an isomorphism
L
:
X
×
M
−→
X
×
M
, and L
−
1
is uniformly bounded for
0
≤
t
≤
1
.
In Theorem 4.11 it is essential that the solution of the saddle point problem
with penalty term is uniformly bounded in
t
for all 0
1. We can think of
the penalty term as a perturbation. It is often supposed to have a stabilizing effect.
Surprisingly, this is not always true, and the norm of the form
c
enters into the
constant in the inf-sup condition. The following example shows that this is not just
an artifact of the proof, which is postponed to the end of this section.
≤
t
≤
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