Civil Engineering Reference
In-Depth Information
As mentioned before, in the V-cycle we distribute the smoothing steps equally
between the phases before and after the coarse-grid correction. Thus, an analogous
symmetric variant is recommended in calculating the starting values.
4.4 Algorithm NI
to compute a starting value v
at the level
≥
0
(symmetric version).
If
=
=
u
0
=
A
−
0
b
0
, and exit the procedure.
0, find
v
0
Let
>
0.
With
v
,
0
0 carry out one step of the multigrid iteration
MGM
, and set
v
,
1
=
=
v
,
0
+
pv
−
1
.
Set
b
−
1
=
p(b
−
A
v
,
1
)
.
Compute an approximate solution
v
−
1
of the equation
A
−
1
u
−
1
=
b
−
1
using
NI
−
1
.
Compute the prolongation of
v
−
1
, and set
v
,
2
=
v
,
1
+
pv
−
1
.
With
v
,
2
carry out one step (in general
p
≥
1 steps) of the multigrid iteration
MGM
, and set
v
=
v
,
3
.
This variant compensates for the effect that in the V-cycle, the oscillating
parts are handled better than the smoother ones. A problem whose smooth parts
are relatively large is given to the block
NI
−
1
, so that the efficiency is increased.
Multigrid Methods with a Small Number of Levels
For many grids it is difficult to carry out more than two levels of coarsening. As
we shall see, however, it still pays to use multigrid methods. In contrast to the
case of a large number of levels, here the solution on the coarsest grid is a major
part of the computational effort. To this effort we have to add the work required
for smoothing, restrictions, and prolongations. This is, of course, less than with a
large number of levels, and has already been estimated in (4.6) above.
For three levels, the number of unknowns on the coarsest grid is ca. 1/16
of that on the finest. In addition, the average bandwidth of the system matrix is
reduced by a factor of about 4. Thus, the amount of work required for the Cholesky
method is reduced by a factor 16
4
2
256. Now if we compute the starting value
with Algorithm 4.1 and add a multigrid cycle, we have to solve
4 systems of equations for the V-cycle,
6 systems of equations for the W-cycle
on the coarsest grid (all with the same matrix). Even if the LU decomposition is
recalculated each time, we still get a saving of a factor of 40 to 64, if we compare
·
=
Search WWH ::
Custom Search