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Ordinary Differential Equations 201
Choosing as test functions the Lagrange polynomials φ := ˜l (n)
i , 1 ≤ i ≤ n − 1, we
obtain the relations −p ′′ n(η (n)
i ) = f(η (n)
i ), 1 ≤ i ≤ n − 1 (see (9.2.15) and (7.4.9)).
To show that pn converges to U when n → +∞, we use theorem 9.4.1. Therefore, we
must bound the error |Fw(φ) − Fw,n(φ)|, φ ∈ P 0 n. Recalling relation (3.8.15) one gets
(9.4.12)
Fw(φ) − Fw,n(φ) = (f − Ĩw,nf,φ)w + [( Ĩw,nf,φ)w − ( Ĩw,nf,φ)w,n]
≤ f − Ĩw,nfwφw
un
+
2 w,n − un2 w
un2 (
w,n
Ĩw,nf,un)w,n
(φ,un)w,n
unw,n unw,n
≤ f − Ĩw,nfw + C |(Ĩw,nf,un)w,n|
φw, ∀φ ∈ P 0 n,
unw,n
where un := P (α,β)
n , n ∈ N, and C > 0 does not grow with n (see section 3.8). By
formula (3.5.1) and orthogonality, one has (Πw,n−1ψ,un)w,n = (Πw,n−1ψ,un)w = 0,
∀ψ ∈ Pn. This implies that
(9.4.13)
≤ γ −1
1
|( Ĩw,nf,un)w,n|
unw,n
≤ Ĩw,nf − Πw,n−1( Ĩw,nf)w,n
Ĩw,nf − fw + f − Πw,n−1fw + Πw,n−1(f − Ĩw,nf)w
, n ≥ 2.
For the last inequality we used (3.8.6) and (2.1.5). By combining (9.4.12) and (9.4.13),
final estimates are obtained using the results of sections 6.2 and 6.6. The results of a
numerical experiment are documented in table 11.2.1.
The study of the approximation of elliptic problems, on the basis of their variational
formulations, is a standard technique in finite element or finite-differences methods. Cel-
ebrated examples are considered for instance in aubin (1972), strang and fix(1973),
ciarlet(1978), fletcher(1984), and in many other books. The simplest approxima-
tion technique is in general represented by the so called Galerkin method. Within the
framework of polynomial approximations, this procedure consists in finding pn ∈ P 0 n
such that