Civil Engineering Reference
In-Depth Information
The fact that transformations to and from shape-regular grids do not generate
extra terms with powers of
h
can also be seen from simple geometric considerations.
Let
F
:
T
1
→
T
2
:
x
→
B x
+
x
0
be a bijective affine mapping. We write
ρ
i
for the
radius of the largest circle inscribed in
T
i
, and
r
i
for the radius of the smallest circle
containing
T
i
.Given
x
∈ R
2
with
x
≤
2
ρ
1
, we find two points
y
1
,z
1
∈
T
1
with
x
=
y
1
−
z
1
, see Fig. 23. Since
F(y
1
), F (z
1
)
∈
T
2
,wehave
Bx
≤
2
r
2
. Thus,
r
2
ρ
1
.
B
≤
(
6
.
9
)
B
−
1
Now exchanging
T
1
and
T
2
, we see that the inverse matrix satisfies
≤
r
1
/ρ
2
,
and thus
r
1
r
2
ρ
1
ρ
2
.
B
·
B
−
1
≤
(
6
.
10
)
Proof of Theorem 6.4.
It suffices to establish the inequality
u
−
I
h
u
m,T
j
≤
ch
t
−
m
for all
u
∈
H
t
(T
j
)
|
u
|
t,T
j
for every triangle
T
j
of a shape-regular triangulation
T
h
.
To
this end, choose a
+
√
2
)
−
1
2
−
1
/
2
reference triangle (5.11) with
2
/
7. Now let
F
:
T
ref
→
T
with
T
=
T
j
∈
T
h
. Applying Lemma 6.2 on the reference triangle
and using the transformation formula in both directions, we obtain
r
=
and
ρ
=
(
2
≥
|
u
−
I
h
u
|
m,T
≤
c
B
−
m
1
/
2
|
det
B
|
|
u
−
I
h
u
|
m,T
ref
≤
c
B
−
m
1
/
2
|
det
B
|
·
c
|
u
|
t,T
ref
≤
c
B
−
m
det
B
|
−
1
/
2
1
/
2
t
|
det
B
|
·
c
B
·|
|
u
|
t,T
B
−
1
)
m
t
−
m
≤
c(
B
·
B
|
u
|
t,T
.
+
√
2
)κ
. Then (6.9)
B
·
B
−
1
By the shape regularity,
r/ρ
≤
κ
, and
≤
(
2
implies
B
≤
h/ ρ
≤
4
h
. Combining these facts, we have
|
u
−
I
h
u
|
,T
≤
ch
t
−
|
u
|
t,T
.
Now squaring and summing over
from 0 to
m
establishes the assertion.
Bilinear Quadrilateral Elements
For quadrilateral elements, we usually use tensor products instead of complete
polynomials. Nevertheless, we can still make use of the techniques developed in
the previous section to establish results on the order of approximation. The simple
but important case of a bilinear element serves as a typical example.
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