Chemistry Reference
In-Depth Information
since
L
h
U
h
f
h
¼
½
40
Thus, we have an identity, and iterations on the coarse grid will not change the
function. This development by Brandt thus satisfies the zero correction at con-
vergence condition. In general,
H
is not zero even when the solution has con-
verged to the exact result, so its inclusion is crucial. The defect correction
t
H
is
a difference between the coarse-grid Laplacian acting on the coarse-grid func-
tion and the restriction of the fine-grid Laplacian acting on the fine-grid func-
tion. It tends to be large in places where the final result changes rapidly; for
example, if we have a single-point charge located at a grid point, the potential
that solves the Poisson equation varies dramatically near that point, and a
peak in
t
H
is observed.
So far we have been concerned with only two grid levels, with grid spa-
cings
h
(fine) and
H
(coarse). We now need one addition to the above discus-
sion. If we use many grid levels, the coarse-grid defect correction on levels two
or more removed from the fine level must include a term that restricts the pre-
vious level's defect correction. Here we label the general coarse-grid level
k
(typically
l
is used then for the finest level):
t
k
L
k
I
k
þ1
u
k
þ1
I
k
þ1
L
k
þ1
u
k
þ1
I
k
þ1
t
k
þ
1
t
¼
þ
½
41
In Eq. [41] we input the current approximation
u
k
for the function on a given
level, which is what is done in the solver as it progresses toward the exact solu-
tion,
U
l
. The coarsest grid level is labeled by the smallest
k.
It is a good exer-
cise to show that Eq. [41] is required on levels two or more removed from the
finest level. Notice that the defect correction on level
k
requires information
only from the next-finer
level.
Once the coarsest level is reached, some iterations are performed there,
and then the solver begins to head back to the fine level (Figure 4). The final
part of the FAS-MG process to be discussed is the
correction
step. Once the
iterations are completed on the coarsest level, the next-finer level is updated
as follows:
ð
k
þ
1
Þ
u
k
þ1
u
k
þ1
I
k
þ
k
ð
u
k
I
k
þ1
u
k
þ1
þ
Þ
½
42
This step interpolates the change in the coarse-grid function during iterations
onto the next-finer level and corrects the current approximation there.
Once corrected, a few iterations are performed on the current level, the
next-finer level is corrected, and the process is repeated until the finest level
is reached. The whole procedure of starting on the fine level, proceeding to
coarser levels, and then correcting and iterating along a path back to the fine
