Civil Engineering Reference
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w
on the reference square [0
,
1]
2
, the derivative
∂
2
w
is
linear in
ξ
. With
φ(ξ)
=
2
ξ
−
1, simple integration gives
For a bilinear function
1
6
[
[0
,
1]
2
φ(ξ)∂
2
ˆ
wdξdη
=
w(
1
,
1
)
−
w(
1
,
0
)
−
w(
0
,
1
)
+
w(
0
,
0
)
]
.
For a bilinear function
w
, affine transformation to a square
T
with edges of length
h
and vertices
a, b, c, d
(in cyclic order) gives
h
6
[
w(a)
−
w(b)
−
w(c)
+
w(d)
]
.
φ∂
2
wdxdy
=
T
2
Here
φ
is a function with
0
,T
=
µ(T )/
3. Repeating this computation for each
square of the partition of
and using (7.6), we get
φ
3
2
q
∗
div
vdx
=
φ∂
2
wdx.
(
7
.
7
)
2
0
Here
=
µ()/
3. With the help of the Cauchy-Schwarz inequality, (7.6) and
(7.7) imply
φ
q
∗
div
vdx
≤
3
2
φ
0
,
∂
2
w
0
,
≤
µ()
1
/
2
v
1
,
4
B
−
1
h
q
∗
0
,
v
1
,
.
≤
In fact,
b(v, q
∗
)
v
1
,
≤
4
B
−
1
h
q
∗
sup
v
∈
X
h
0
,
.
(
7
.
8
)
Thus, the inf-sup condition only holds for some constant depending on
h
. This
clearly shows that
we cannot check the inf-sup condition by merely counting degrees
of freedom and using dimensional arguments.
In order to verify the Brezzi condition with a constant independent of
h
,
we have to further restrict the space
R
h
. This can be done by combining four
neighboring squares into a macro-element. The functions sketched in Fig. 40 form
a basis on the level of the macro-elements for the functions which are constant on
every small square.
If we eliminate those functions in each macro-element which correspond to
the pattern in Fig. 40d, we get the desired stability independent of
h
; cf. Girault
and Raviart [1986], p. 167 or Johnson and Pitkaranta [1982]. However, in doing
so, we lose much of the simplicity of the original approximations. Therefore, the
stabilized
Q
1
-
P
0
elements are not considered to be competitive.
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