Civil Engineering Reference
In-Depth Information
Proof.
Let
v
h
∈
S
h
. For convenience, set
u
h
−
v
h
=
w
h
. Then by the uniform
continuity and (1.2)-(1.3), we have
α
2
u
h
−
v
h
≤
a
h
(u
h
−
v
h
,u
h
−
v
h
)
=
a
h
(u
h
−
v
h
,w
h
)
=
a(u
−
v
h
,w
h
)
+
[
a(v
h
,w
h
)
−
a
h
(v
h
,w
h
)
]
+
[
a
h
(u
h
,w
h
)
−
a(u, w
h
)
]
=
a(u
−
v
h
,w
h
)
+
[
a(v
h
,w
h
)
−
a
h
(v
h
,w
h
)
]
−
[
, w
h
−
h
,w
h
]
.
Dividing through by
u
h
−
v
h
=
w
h
and using the continuity of
a
,weget
u
h
−
v
h
≤
C
.
u
−
v
h
+
|
a(v
h
,w
h
)
−
a
h
(v
h
,w
h
)
|
+
|
h
,w
h
−
, w
h
|
w
h
w
h
Since
v
h
is an arbitrary element in
S
h
, the assertion follows from the triangle
inequality
−
u
h
≤
−
v
h
+
u
h
−
v
h
u
u
.
Dropping the conformity condition
S
h
⊂
V
has several consequences. In
particular, the
H
m
-norm might not be defined for all elements in
S
h
, and we have
to use mesh-dependent norms
·
h
as discussed, e.g., in II.6.1.
We assume that the bilinear forms
a
h
are defined for functions in
V
and in
S
h
, and that we have ellipticity and continuity:
a
h
(v, v)
2
h
≥
α
v
for all
v
∈
S
h
,
(
1
.
4
)
|
a
h
(u, v)
|≤
C
u
h
v
h
for all
u
∈
V
+
S
h
,v
∈
S
h
,
with some positive constants
α
and
C
independent of
h
.
The following lemma is often denoted as the
second lemma of Strang.
1.2 Lemma of Berger, Scott, and Strang.
Under the above hypotheses there
exists a constant c independent of h such that
u
−
u
h
h
≤
c
inf
.
|
a
h
(u, w
h
)
−
h
,w
h
|
w
h
h
v
h
∈
S
h
u
−
v
h
h
+
sup
w
h
∈
S
h
Remark.
The first term is called the
approximation error
, and the second one is
called the
consistency error
.
Proof.
Let
v
h
∈
S
h
. From (1.4) we see that
α
u
h
−
v
h
2
h
≤
a
h
(u
h
−
v
h
,u
h
−
v
h
)
=
a
h
(u
−
v
h
,u
h
−
v
h
)
+
[
h
,u
h
−
v
h
−
a
h
(u, u
h
−
v
h
)
]
.
Dividing by
u
h
−
v
h
h
and replacing
u
h
−
v
h
by
w
h
,wehave
u
h
−
v
h
h
≤
α
−
1
C
u
−
v
h
h
+
|
a
h
(u, w
h
)
−
h
,w
h
|
.
w
h
h
The assertion now follows from the triangle inequality as in the proof of the first
lemma.
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