Untitled - Cdm.unimo.it

Ordinary Differential Equations 197
(9.3.25)
⎧
⎨ ∀ψ ∈ X, ∃φ ∈ Y : |B(ψ,φ)| ≥ C2 ψXφY,
⎩
∀φ ∈ Y φ ≡ 0, ∃ψ ∈ X : B(ψ,φ) = 0.
Extensions of this kind are studied, for instance, in nečas(1962). This approach was
used in canuto and quarteroni(1984) to provide an alternative variational formula-
tion to problem (9.1.5).
9.4 Approximation of problems in the weak form
Let us see how to use the results of the previous section to carry out the convergence
analysis for the approximation schemes (9.2.14) and (9.2.15). If Bw and Fw are
respectively given by (9.3.2) and (9.3.3) ( X ≡ H 1 0,w(I)), we already noticed that (9.3.4)
has a unique weak solution and generalizes problem (9.1.4) with σ1 = σ2 = 0. We also
write (9.2.14) in a variational form.
For any n ≥ 2, we consider the finite dimensional space P 0 n ⊂ X (see (6.4.11)). We
remark that P 0 n is a Hilbert space with the same inner product defined in X. Then,
we introduce the linear operator Fw,n : P 0 n → R as follows:
(9.4.1) Fw,n(φ) :=
where f ∈ L 2 w(I).
(Πw,n−2f)φw dx, ∀φ ∈ P
I
0 n,
For the sake of simplicity, let w be the ultraspherical weight function with ν := α = β
satisfying −1 < ν < 1. With the help of theorem 9.3.2, we can apply theorem 9.3.1 to
obtain a unique solution pn ∈ P 0 n of the problem
(9.4.2) Bw(pn,φ) = Fw,n(φ), ∀φ ∈ P 0 n,