Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 260

Problems

u h − u ≤ ch 2
for some constant c> 0. In addition, suppose the convergence rate for the two-grid
method with ν = ν 2 post-smoothings satisfies ρ< 1 / 10.
Suppose a user (who perhaps is not familiar with the multigrid method) ap-
plies a classical relaxation method, starting with the solution on the 2 h grid. Show
that after ν steps,

4.6 Suppose the finite element approximation u h ∈ S h is such that

3

2 c · h 2 .
Why doesn't a corresponding assertion hold for more than two grids?

u h,ν − u ≤

4.7 Compare the operation counts of
(a) NI with the V-cycle,
(b) NI with the W-cycle,
(c) the symmetric version of NI with the V-cycle.

4.8 Suppose we want to insert a so-called F-cycle (see Fig. 55) between the
V-cycle and the W-cycle as follows:
For = 2, the F-cycle and W-cycle coincide.

1.
Find the recurrence formula analogous to (3.17), and determine the rates numeri-
cally for c =

For ≥ 3, perform both an F-cycle and a V-cycle at level −

1

1 ,c =

2 ,ν =

2, and ≤ 8.

◦ ◦ 4

◦ ◦ ◦

3

◦ ◦ ◦ ◦ ◦ 2

◦◦◦◦◦ ◦◦

1

•• • •

0

Fig. 55. F-cycle on five levels

4.9 Compare nested iteration NI using inner V-cycles with the F-cycle of MGM .
What is the difference?