Untitled - Cdm.unimo.it

180 Polynomial Approximation of Differential Equations
in (7.4.9). This is obtained starting from the set of grid-points η (n)
j , 0 ≤ j ≤ n (finegrid).
Another set of points in [−1,1] (coarse-grid) is given by the nodes η (k)
j , 0 ≤ j ≤ k,
where 2 ≤ k ≤ n. A polynomial p ∈ Pn, determined by its values on the fine-grid,
is evaluated at the coarse-grid by using formula (3.2.7). On the other hand, we can go
from the coarse-grid to the fine-grid by extrapolation, i.e.
k
(8.7.1) p(η (n)
i ) =
j=0
p(η (k)
j ) ˜ l (k)
j (η (n)
i ), 0 ≤ i ≤ n.
In short, the philosophy of the multigrid method is the following. Since the eigenvalues
of D (n) are not available (the cost of their computation is very high), an alternative
to varying the parameter θ is to accelerate the convergence by using different grids.
A good treatment of low modes is obtained by applying the Richardson method to
the matrix D (k) (k small) defined on the coarse-grid. Then, we can switch to the
fine-grid by (8.7.1), and work with the matrix D (n) to damp the high modes with
a relatively small θ. In general, the algorithm applies to different sets of grid-points,
selected in turn, according to some prescribed rule (the most classical one is the V-
cycle). There are many ways to advance the method. These depend on the problem
and the competence of the user. A comparative analysis is given in heinrichs(1988).
Other results are provided in muñoz(1990). Early applications within the framework of
spectral methods, for more specialized problems, are examined, for instance, in zang,
wong and hussaini(1982), streett, zang and hussaini(1985), brandt, fulton
and taylor(1985), zang and hussaini(1986). For a general overview of the method,
we refer to hackbusch(1985).
Using the multigrid method for the preconditioned systems in place of a plain
preconditioned iterative scheme, does not improve very much the rate of convergence
for problems in one dimension. More interesting are the applications in two or three
dimensions. Performances also depend on the velocity and the cost to transfer the data
from one grid to the next. Particular efficiency is obtained in the Chebyshev case. When
n is even and k = n/2, recalling relation (3.8.16), all the points of the coarse-grid are
also points of the fine-grid. Furthermore, the inverse formula (8.7.1) is implemented via
FFT.