Civil Engineering Reference
In-Depth Information
§ 2. Convergence of Multigrid Methods
A multigrid method is said to have
multigrid convergence
if the error is reduced
by a factor of at least
ρ<
1 in each iteration cycle, where
ρ
is
independent of h
.
In this case the convergence as
h
→
0 cannot be arbitrarily slow, in contrast to
classical iterative methods. The factor
ρ
is called the
convergence rate
. Clearly, it
is a measure of the speed of convergence.
Independently from Fedorenko [1961], Hackbusch [1976] and Nicolaides
[1977] also presented convergence proofs. Here we make use of an idea employed
by Bank and Dupont [1981] in their proof. A general framework due to Hackbusch
[1985] admits to break convergence proofs into two separate parts. In this way they
become very transparent. A
smoothing property
1
ν
γ
v
h
Y
ν
v
h
X
≤
ch
−
β
S
(
2
.
1
)
is combined with an
approximation property
v
h
−
u
2
h
Y
≤
ch
β
v
h
X
,
(
2
.
2
)
where
u
2
h
is the coarse-grid approximation of
v
h
. Then for large
ν
, the product
of the two factors is smaller than 1 and independent of
h
. In particular, it follows
that the convergence rate tends to zero for large numbers of smoothing steps.
The various proofs differ in the choice of the norms
·
X
and
·
Y
, where
·
X
generates a stronger topology than
·
Y
. The pair (2.1) and (2.2) have to fit
together in exactly the same way as the approximation property (II.6.20) and the
inverse estimate (II.6.21). It is clear that we need two norms, or more generally
two measures, for specifying the error. In addition to measuring the
size
of the
error (w.r.t. whichever norm), we also have to measure the
smoothness
of the error
function.
It is the goal of this section to establish convergence of the two-level iteration
under the following hypotheses:
2.1 Hypotheses.
(1) The boundary-value problem is
H
1
-or
H
0
-elliptic.
(2) The boundary-value problem is
H
2
-regular.
(3) The spaces
S
belong to a family of conforming finite elements with uniform
triangulations, and the spaces are nested, i.e.,
S
−
1
⊂
S
.
(4) We use nodal bases.
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