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184 Polynomial Approximation of Differential Equations
(9.2.2) U(x) − pn(x) =
x
−1
(f − Πw,n−1f)(s) ds, ∀x ∈ Ī.
At this point, an estimate of the rate of convergence to zero of the error U −pn, n ≥ 1,
in some suitable norm, is straightforward after recalling the results of section 6.2. For
example, when −1 < α < 1 and −1 < β < 1, it is easy to obtain uniform convergence
with the help of the Schwarz inequality and of theorem 6.2.4. Indeed, for k ∈ N, one
has
(9.2.3) sup
x∈[−1,1]
|U(x) − pn(x)| ≤ C
k 1
fH k
w n
(I), ∀n > k.
The space H k w(I), k ∈ N, is defined in (5.7.1). As already observed in chapter six, the
more regular the function f, the faster the decay of the error is.
Another approach is to construct the approximation scheme in the physical space.
This leads to the collocation (or pseudospectral) method. Now, the approximating poly-
nomials pn ∈ Pn, n ≥ 1, satisfy
⎧
⎨
(9.2.4)
⎩
p ′ n(η (n)
i ) = f(η (n)
i ) 1 ≤ i ≤ n,
pn(η (n)
0 ) = σ.
Here, we impose the boundary condition at x = −1 and we collocate the differential
equation at the nodes η (n)
i , 1 ≤ i ≤ n. We are left with a linear system in the unknowns
pn(η (n)
j ), 0 ≤ j ≤ n. The corresponding matrix is obtained following the guideline of
section 7.4. We remark that the computed values pn(η (n)
j ), 0 ≤ j ≤ n, do not coincide
in general with the values U(η (n)
j ), 0 ≤ j ≤ n.
We can study the convergence of the collocation method by arguing as follows. We
define r(x) := 1 + x, x ∈ Ī. Then, we note that (9.2.4) is equivalent to
(9.2.5)
⎧
⎨p
⎩
′ n = [ Ĩw,n(fr)]r −1 pn(−1) = σ,
in ] − 1,1],