Untitled - Cdm.unimo.it

200 Polynomial Approximation of Differential Equations
≤ C −1
2 sup
φ∈Xn
φ≡0
B(χn − pn,φ)
φX
= C −1
2 sup
φ∈Xn
φ≡0
|F(φ) − Fn(φ)|
, n ∈ N.
φX
Inequality (9.4.4) follows by combining (9.4.6), (9.4.7) and (9.4.8).
Returning to problem (9.4.2), we apply the previous theorem to get the estimate
(9.4.9) U −pnH1 0,w (I) ≤ C
inf
χ∈P0 U −χH1 0,w (I) + f −Πw,n−2fL2 w (I)
n
, ∀n ≥ 2.
We further bound this error using the results of chapter six. We can consider inequalities
(6.4.15) and (6.4.6) after taking χ := Π 1 0,w,nU in (9.4.9). We remark that f = −U ′′ in
I. Thus, we finally get
(9.4.10) U − pn H 1 0,w (I) ≤ C
k 1
fH k
w n
(I), ∀n > k ≥ 0.
This shows that pn converges to U with a rate depending on the regularity of f. Due
to inequality (5.7.4), this convergence is uniform.
We remark that the function χn, n ≥ 2, defined in the proof of theorem 9.4.1, here
coincides with the polynomial ˆ Π 1 0,w,nU ∈ P 0 n, introduced in (6.4.13). The existence of
such a polynomial follows from theorem 9.3.1, and estimates of the error U − ˆ Π 1 0,w,nU
can be recovered from (9.4.7).
The same kind of analysis applies to the collocation method (9.2.15) in the case
σ1 = σ2 = 0. It is sufficient to recall the quadrature formula (3.5.1) and modify (9.4.1)
according to
(9.4.11) Fw,n(φ) :=
n
j=0
f(η (n)
j )φ(η (n)
j ) ˜w (n)
j , ∀φ ∈ P 0 n,
where f ∈ C0 ( Ī). We recall the Schwarz inequality and theorem 3.8.2 to check (9.3.7).
Actually, one has |Fw,n(φ)| ≤ C3f C 0 (Ī)φ H 1 0,w (I), ∀φ ∈ P 0 n, ∀n ≥ 2.