Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 254

In the multigrid method, the low frequency parts are handled more efficiently
in the W-cycle than in the V-cycle. Hence, it is not surprising that in (3.18) the
maximum is attained for β =

1. Thus, it makes sense to insert W-cycles once in
a while if the number of levels is very large.
The proof shows that for large ν , the contraction number decreases only like
ν − 1 / 2 .

If both pre-smoothing and post-smoothing are used, the rate of decrease is ν − 1 ,
as shown by the duality technique of Braess and Hackbusch [1983]. We remark
that later convergence proofs for the V-cycle usually make use of an algebraic hy-
pothesis instead of the H 2 -regularity; cf. Bramble, Pasciak, Wang, and Xu [1991].
The question of whether appropriate algebraic hypotheses are really independent
of H 2 -regularity remains open, despite the paper of Parter [1987].
If the regularity hypothesis 2.1(2) is not satisfied, we have to expect a less
favorable convergence rate. Then as suggested by Bank, Dupont, and Yserentant
[1988], it makes more sense to use the multigrid method as a preconditioner for
a CG method rather than as a stand-alone iteration. In fact these authors go one
step further, and use the multigrid idea only for the construction of a so-called
hierarchical basis . Then the convergence rate behaves like 1

h ) − p ) ,

1

− O (( log

which is still quite good for practical computations.

Problems

3.9 Show that for the W-cycle,

provided ρ 1 < 1

ρ 1

ρ 2 sup ≤ ,

2 . − ρ 1

1

− ρ 2

Hint: Use (3.9) to derive a recurrence formula for 1

.

3.10 Show that for large c , the recurrence formula (3.17) gives the two-grid rate

1

c 2 ) ν ,

ρ 1 ≤ ( 1 −

and compare with (3.11).

3.11 The amount of computation required for the W-cycle is approximately 50 %
larger than for the V-cycle. Use the tables to compare the error reduction of three
V-cycles and two W-cycles.