Civil Engineering Reference
In-Depth Information
see Theorem A.1 in Vogelius [1983]. Following the usual procedure (see, e.g.,
Theorem III.4.5) we have
2
µ(ε(u
h
−
+
(
div
v, p
h
−
=
∈
P
h
u), ε(v))
0
Q
h
p)
0
u
,v
for all
v
X
h
,
−
λ
−
1
(p
h
−
Q
h
p, q)
0
for all
q
∈
M
h
.
The functionals
u
and
p
can be expressed in terms of
u
−
P
h
u
and
p
−
Q
h
p
. From
the estimates (3.32) and (3.33) we immediately obtain
(
div
(u
h
−
P
h
u), q)
0
=
p
,q
u
−
1
+
p
0
≤
ch
f
0
.
Hence,
u
−
u
h
1
+
λ
div
u
−
p
h
0
≤
ch
f
0
, (
3
.
34
)
where
c
is a constant independent of
λ
.
The finite element method (3.31)
h
is robust.
Nonconforming methods are also very popular for treating nearly incom-
pressible materials. The finite element discretization (3.31)
h
of the saddle point
problem can also be interpreted as a nonconforming method, and this discretiza-
tion will serve as a model for analyzing nonconforming methods for the problem
with a small parameter.
H
1
3.9 Remark.
L
2
() be the solution of
the finite element discretization (3.31)
h
. Define a discrete divergence operator by
div
h
:
H
1
()
→
M
h
(
div
h
v, q)
0
=
(
div
v, q)
0
Let u
h
∈
X
h
⊂
() and p
h
∈
M
h
⊂
(
3
.
35
)
for all q
∈
M
h
.
Then u
h
is also the solution of the variational problem
2
µ
ε(v)
2
0
2
0
+
λ
div
h
v
−
, v
−→
min
v
∈
X
h
!
(
3
.
36
)
Indeed, the solution
u
h
of (3.36) is characterized by
2
µ(ε(u
h
), ε(v))
0
+
λ(
div
h
u
h
,
div
h
v)
0
=
, v
for all
v
∈
X
h
.
Setting
p
h
:
=
λ
div
h
u
h
by analogy to (3.30), we have
2
µ(ε(u
h
), ε(v))
0
+
(
div
h
v, p
h
)
0
=
, v
for all
v
∈
X
h
,
λ
−
1
(p
h
,q)
0
M
h
.
Here the operator div
h
is met only in inner products with the other factor in
M
h
.
Now, from the definition (3.35) we know that the operator div
h
may be replaced
here by div. Therefore,
u
h
together with
p
h
:
(
div
h
u
h
,q)
0
−
=
0
for all
q
∈
λ
div
h
u
h
satisfy (3.31)
h
.
We turn to the analysis of (3.36). The inequality (3.32)
2
implies the estimate
=
div
v
1
, but in most of the nonconforming methods there
is only a weaker estimate available. Moreover, the discrete divergence does not
result from an orthogonal projection in many cases. A basis for an alternative is
offered by the following regularity result in Theorem 3.1 of Arnold, Scott, and
Vogelius [1989].
Given u with
div
u
∈
H
1
(), there exists w
∈
H
2
()
∩
H
0
()
such that
div
v
−
div
h
v
0
≤
ch
div
w
=
div
u
and
w
2
≤
c
div
u
1
.
(
3
.
37
)
Search WWH ::
Custom Search