Civil Engineering Reference
In-Depth Information
An Alternative Short Proof
O
(ν
−
1
/
2
)
, there is a much
shorter proof. Then we do not need to introduce the scale of discrete norms (2.3),
but the smoothing property of the matrix operations is less transparent without
this scale. We refer to that proof for completeness; we need only rearrange some
results known from the previous investigation.
The smoothing property and the approximation property also hold with
If we are content with a convergence rate which is
1
2
.
·
X
=·
1
,
·
Y
=·
0
,β
=
1
,
and
γ
=
Indeed, the approximation property (2.13)
v
−
u
2
h
0
≤
ch
v
−
u
2
h
1
≤
ch
v
1
is immediate from the Aubin-Nitsche lemma. Moreover, when we apply Lemma
2.4 to the case
s
=
0,
s
+
t
=
1, we may prove the smoothing property
x
ν
1
≤
cν
−
1
/
2
x
0
0
without reference to the discrete norms. The rest of proof proceeds as for Theorem
2.9.
Some Variants
It is easy to see the connection with the somewhat different terminology of Hack-
busch [1985]. The matrix representation of the two-grid iteration is
u
1
,k
+
1
−
u
1
=
M(u
1
,k
−
u
1
)
(
2
.
19
)
with
pA
−
1
ν
2
(I
ν
1
M
=
S
−
2
h
rA
h
)
S
(
2
.
20
)
ν
2
(A
−
1
−
pA
−
1
ν
1
.
=
S
2
h
r)A
h
S
h
In particular,
pA
−
1
2
h
rA
h
describes the coarse-grid correction for a two-grid method.
Writing the smoothness and approximation properties in the form
c
ν
ν
h
−
2
,
(A
−
1
−
pA
−
1
2
h
r)
≤
ch
2
A
h
S
≤
(
2
.
21
)
h
M
≤
c/ν <
1 for suffi-
ciently large
ν
. Since
|||
Ax
|||
0
= |||
x
|||
2
, the smoothing and approximation proper-
ties follow from (2.21) with the same norms and parameters as in (2.15).
The smoothing property is usually established as in Lemma 2.4 whenever the
convergence of the multigrid method is carried out in Hilbert spaces. The proof of
convergence w.r.t. the maximum norm by Reusken [1992] is different.
and using
S
≤
1, we get the contraction property
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