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Ordinary Differential Equations 185
where Ĩw,n, n ≥ 1, is the interpolation operator at the points η (n)
j , 0 ≤ j ≤ n (see
section 3.3). It turns out that [ Ĩw,n(fr)]r −1 is the interpolant of f at the nodes
η (n)
j , 1 ≤ j ≤ n. This shows the equivalence of (9.2.5) with (9.2.4). The analysis of the
error U −pn, n ≥ 1, is then recovered from the estimates of the error [fr− Ĩw,n(fr)]r −1 .
These results have been developed in section 6.6.
The two approximation methods described here differ in the treatment of the func-
tion f. Projection and interpolation operators have been used respectively. A small
difference is due to the aliasing error (see section 4.2), but the approximating solutions
behave similarly. The first approach is more effective in terms of computational cost.
Tau and collocation methods are respectively used to approximate the solution of
problem (9.1.3) according to the following schemes:
(9.2.6)
(9.2.7)
⎧
⎨
⎩
⎧
⎨
p ′ n + Πw,n−1(Apn) = Πw,n−1f in ] − 1,1],
⎩
pn(−1) = σ ,
p ′ n(η (n)
i ) + A(η (n)
i )pn(η (n)
i ) = f(η (n)
i ) 1 ≤ i ≤ n,
pn(η (n)
0 ) = σ ,
pn ∈ Pn, n ≥ 1.
The corresponding linear systems are given by (7.3.3) and (7.4.7). Here, the second
approach is preferable, having the coefficients of the matrix relative to the tau method
a complicate expression, for a non-constant function A.
The analysis of convergence is now more involved and general results are not avail-
able. We examine for instance the collocation method in the Legendre case. We recall
that the norm in the space L 2 w(I) is defined in (5.2.4).