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i.e.,
B
−
A
is assumed to be positive semidefinite and tacitly
B
is assumed to
be symmetric. Often only the weaker condition
A
≤
ωB
with
ω<
2 is required,
but we prefer to have the assumption without an extra factor in order to avoid some
inconvenient factors in the estimates. Some standard techniques for dealing with
the approximate solution above are found in Problems IV.4.14-17.
We recall (5.11) and following the standard notation, we define the linear
mapping
=
B
−
1
Q
A
=
B
−
1
A
P
.
T
:
(
5
.
14
)
From (5.12) we know that the correction of
u
in the subspace
S
yields the new
iterate
u
+
T
(u
−
u)
, and its error is
(I
−
T
)(u
−
u).
We consider the multigrid
V
-cycle with post-smoothing only. Consequently the
error propagation operator for one complete cycle is
E
:
=
E
L
where
E
:
=
(I
−
T
)(I
−
T
−
1
)...(I
−
T
0
),
=
0
,
1
,...,L,
(
5
.
15
)
=
I
. This representation elucidates that the subspace corrections are
applied in a multiplicative way.
and
E
−
1
:
Assumptions
The assumptions refer to the family of finite element spaces
S
and the comple-
mentary spaces
W
specified in (5.7).
Assumption A1.
There exists a constant
K
1
such that for all
v
∈
W
,
=
0
,
1
,...,L
,
L
L
2
.
(B
v
,v
)
≤
K
1
v
(
5
.
16
)
=
0
=
0
AssumptionA2
(Strengthened Cauchy-Schwarz Inequality).
There exist constants
γ
k
=
γ
k
with
a(v
k
,w
)
≤
γ
k
(B
k
v
k
,v
k
)
1
/
2
(B
w
,w
)
1
/
2
for all
v
k
∈
S
k
,w
∈
W
(
5
.
17
)
if
k
≤
. Moreover, there is a constant
K
2
such that
γ
k
x
k
y
≤
K
2
x
k
1
/
2
y
2
1
/
2
L
L
L
L
+
1
.
for
x,y
∈ R
(
5
.
18
)
k,l
=
0
k
=
0
=
0
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