Civil Engineering Reference
In-Depth Information
The upper bound (8.12) means that the estimator
η
R
is
reliable
and the lower
bound (8.13) that it is also
efficient
.
Proof of the upper estimate (8.12).
We start by using a duality argument to find
|
u
−
u
h
|
1
=
sup
(
∇
(u
−
u
h
),
∇
w)
0
.
(
8
.
14
)
H
0
|
w
|
1
=
1
,w
∈
We make use of the following formula which also appeared in establishing the
Cea Lemma:
S
h
. (
8
.
15
)
We now consider the functional
corresponding to (8.14), apply Green's formula,
and insert the residuals (8.2) and (8.3):
(
∇
(u
−
u
h
),
∇
v
h
)
0
=
0
for
v
h
∈
, w
:
=
(
∇
(u
−
u
h
),
∇
w)
0
,
=
(f, w)
0
,
−
(
∇
u
h
,
∇
w)
0
,T
T
=
(f, w)
0
,
−
(
−
u
h
,w)
0
,T
+
(
∇
u
h
·
n, w)
0
,e
T
e
⊂
∂T
∂u
h
∂n
,w
0
,e
=
(u
h
+
f, w)
0
,T
+
T
e
⊂
h
=
(R
T
,w)
0
,T
+
(R
e
,w)
0
,e
.
(
8
.
16
)
T
e
⊂
h
By Clement's results on approximation, cf. II.6.9, for given
w
∈
H
0
()
there
exists an element
I
h
w
∈
S
h
with
w
−
I
h
w
0
,T
≤
ch
T
∇
w
0
, ω
T
for all
T
∈
T
h
,
(
8
.
17
)
≤
ch
1
/
2
w
−
I
h
w
∇
w
0
, ω
T
for all
e
⊂
h
.
(
8
.
18
)
0
,e
e
Here
ω
T
is the neighborhood of
T
specified in (II.6.14) which is larger than
ω
T
.
Since the triangulations are assumed to be shape regular,
{
ω
T
;
T
∈
T
h
}
covers
only a finite number of times. Hence, (8.15) implies
, w
=
, w
−
I
h
w
e
⊂
h
≤
R
T
w
−
I
h
w
+
R
e
w
−
I
h
w
0
,T
0
,T
0
,e
0
,e
T
≤
c
T
h
T
R
T
0
,T
|
w
|
1
,T
+
c
e
(
8
.
19
)
h
1
/
2
R
e
0
,e
|
w
|
1
,ω
e
e
⊂
h
≤
c
T
η
T,R
|
w
|
1
,T
≤
cη
R
|
w
|
1
,
.
The last inequality follows from the Cauchy-Schwarz inequality for finite sums.
Combining (8.18) and (8.19) with Friedrichs' inequality and the duality argument
(8.14), we get the global upper error bound (8.12).
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