Chemistry Reference
In-Depth Information
In this last formula the coarse-grid index
j
is located at the point just to the left
of the fine-grid point (Figure 3). In higher dimensions, the best way to enact
the interpolation is to perform the operation as a series of interpolations on
lines, that is, as a series of one-dimensional operations. This has the same
approximation order and is easy to code. The symbol used for the interpola-
tion operator is
I
H
.
Now that we have the two transfer operations in hand, how do we
design the coarse-grid problem? Consider again the matrix representation of
the Poisson equation:
L
h
U
h
f
h
¼
½
34
Here we use the uppercase
U
h
to represent the exact numerical solution on the
fine grid. We now pass that exact solution to the coarse grid by restriction and
examine the same type of equation:
L
H
u
H
f
H
¼
½
35
where
u
H
I
h
U
h
¼
½
36
and
f
H
I
h
f
h
¼
½
37
The difficulty here is that
u
H
does not solve the coarse-grid equation (which is
why lowercase was used for this function above). This situation immediately
becomes a problem because once we have the exact solution on the fine grid,
nothing should happen on the coarse grid. Brandt introduced a clever solution
to this problem called the FAS method.
107,108
We modify the coarse-grid equation (Eq. [35]), with an added function
t
H
, the
defect correction
, defined as follows (assuming we have the exact fine-
grid solution
U
h
for now—of course, we do not have that function during the
solution process, but this argument illustrates a key point):
H
L
H
I
h
U
h
I
h
L
h
U
h
t
¼
½
38
This term is added to the right side of the coarse-grid equation [35] to obtain
L
H
U
H
f
H
H
I
h
f
h
L
H
I
h
U
h
I
h
L
h
U
h
L
H
I
h
U
h
¼
þ
t
¼
þ
¼
½
39
