Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 231

(2) For more than two levels, it makes a difference whether we perform the
smoothing before or after the coarse-grid correction. This is controlled by the pa-
rameters ν 1 and ν 2 . For simplicity, frequently only the pre-smoothing is performed.
However, for the V-cycle, it is better to do an equal amount of smoothing both
before and after, i.e. to choose ν 1 = ν 2 .
(3) Choosing µ =

2 leads to either a V-cycle or a W-cycle . The
reason for this terminology is clear from the shape of the corresponding schemes
shown in Fig. 50. Obviously a W-cycle is more expensive than a V-cycle.
In order to ensure that in running through several levels the error does not
build up too much, in the early use of the multigrid method most people chose
W-cycles. However, most problems are so well-behaved that multigrid algorithms
with the V-cycle are faster. (For more than four levels, it is better to insert one
W-cycle after every three V-cycles; see Problem 3.12.)
(4) We solve the system of equations corresponding to the variational problem
on the coarsest grid using Gauss elimination or some other direct method.
(5) In practice, an auxiliary grid can be so coarse that it would never be used
as the final grid. For the Poisson equation on the unit square, it is even possible that
the grid is coarsened so much that the coarsest grid contains only one (interior)
point. This does not ruin the convergence rate of multigrid algorithms.

1or µ =

◦ ◦ ◦ ◦

2

◦◦ ◦◦◦

1

• ••

0

a)

◦ ◦ ◦ ◦ 3

◦ ◦ ◦ ◦ ◦

2

◦◦ ◦◦◦◦◦◦ 1

• •• •• 0

b)

Fig. 50. V-cycle and W-cycle on three and four levels