Untitled - Cdm.unimo.it

186 Polynomial Approximation of Differential Equations
Theorem 9.2.1 - Let α = β = 0 and A : Ī → R, be a continuous positive function
satisfying A(x) ≥ ǫ, ∀x ∈ Ī, for a given ǫ > 0. Then
(9.2.8) lim
n→+∞ pn − U L 2 w (I) = 0,
where pn, n ≥ 1, is the solution of (9.2.7) and U is the solution of (9.1.3).
Proof - We first note that, by the triangle inequality (2.1.5), one has
(9.2.9) pn − U L 2 w (I) ≤ pn − Ĩw,nU L 2 w (I) + Ĩw,nU − U L 2 w (I), n ≥ 1.
The last term of the right-hand side tends to zero in view of theorem 6.6.1. After defining
sn := pn − Ĩw,nU ∈ Pn, n ≥ 1, we require an estimate for the error sn L 2 w (I), n ≥ 1.
Equations (9.1.3) and (9.2.7) yield
(9.2.10)
⎧
⎨
⎩
s ′ n(η (n)
i ) + A(η (n)
i )sn(η (n)
i ) = (U − Ĩw,nU) ′ (η (n)
i ) 1 ≤ i ≤ n,
sn(η (n)
0 ) = 0.
Using the quadrature formula (3.5.1) with w ≡ 1, we obtain
(9.2.11)
Next, we observe that
=
s
I
′ nsn dx +
n
i=0
n
i=0
tions on A and the Schwarz inequality, one gets
A(η (n)
i )s 2 n(η (n)
i ) ˜w (n)
i
(U − Ĩw,nU) ′ (η (n)
i ) sn(η (n)
i ) ˜w (n)
i , n ≥ 1.
I s′ nsndx = 1
2 s2 n(1) ≥ 0, n ≥ 1. Therefore, from the assump-
(9.2.12) ǫ snw,n ≤ Ĩw,nU ′ − ( Ĩw,nU) ′ w,n, n ≥ 1,
where we note that Ĩw,nU ′ coincides with U ′ at the nodes. The norm ·w,n, n ≥ 1, is
given in (3.8.2). By virtue of theorem 3.8.2, this norm is equivalent to the L 2 w(I) norm.
Besides, when U ∈ C1 ( Ī), the right-hand side of (9.2.12) tends to zero. Actually, by
(3.8.6), one has