Civil Engineering Reference
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and thus
∇
1
q
and
∇
2
q
vanish at all interior nodes. This happens if
q
i
+
1
/
2
,j
+
1
/
2
=
q
i
−
1
/
2
,j
−
1
/
2
,
i
+
1
/
2
,j
−
1
/
2
=
q
i
−
1
/
2
,j
+
1
/
2
.
These equations do not mean that
q
must be a constant. They only require that
a
for
i
+
j
even,
q
i
+
1
/
2
,j
+
1
/
2
=
b
for
i
+
j
odd.
Here the numbers
a
and
b
must be chosen so that (6.3) holds, and thus
q
∈
L
2
,
0
()
.
In particular,
a
and
b
must have opposite signs, giving the
checkerboard pattern
shown in Fig. 38. In the following we use
ρ
to denote the corresponding pressure
(up to a constant factor).
•
•
•
•
•
•
•
•
•
•
•
+ − + − + − + − + −
•
•
•
•
•
•
•
•
•
•
•
− + − + − + − + − +
•
•
•
•
•
•
•
•
•
•
•
+ − + − + − + − + −
•
•
•
•
•
•
•
•
•
•
•
− + − + − + − + − +
•
•
•
•
•
•
•
•
•
•
•
− + − + − + −
•
•
•
•
•
•
•
•
+ − + − + − +
•
•
•
•
•
•
•
•
− + − + − + −
•
•
•
•
•
•
•
•
Fig. 38.
Checkerboard instability
7.1 Remark.
The inf-sup condition is an
analytic
property, and should not be
interpreted just as a purely
algebraic
one. This fact becomes clear from the mod-
ification of
Q
1
-
P
0
elements needed to achieve stability. We start with a reduction
of the space
M
h
so that the kernel of
B
h
becomes trivial. Since
is assumed to
be connected, ker
B
h
=
span[
ρ
] has dimension 1. The mapping
B
h
R
h
−→
X
h
:
is injective on the space
=
ρ
⊥
={
q
∈
M
h
;
(q, ρ)
0
,
=
R
h
:
0
}
.
Unfortunately this is not sufficient for full stability.
There is a constant
β
1
>
0 such that
b(v, q)
v
sup
v
1
≥
β
1
h
q
0
for
q
∈
R
h
(
7
.
3
)
∈
X
h
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