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198 Polynomial Approximation of Differential Equations
where Bw has been introduced in (9.3.2). We note that the space in which we seek
the solution is rather small now. On the other hand, we require (9.4.2) to be satisfied
only for test functions belonging to a subspace of X.
It is very easy to check that pn is solution to (9.2.14) where σ1 = σ2 = 0. In fact,
setting a(x) := (1 −x 2 )w(x), x ∈ I, and recalling that p ′′ n +Πw,n−2f ∈ Pn−2, one has
−
p
I
′′ nφw dx = Fw,n(φ), ∀φ ∈ P 0 n
− p
I
′′ nχa dx =
⇕
(Πw,n−2f)χa dx, ∀χ ∈ Pn−2
I
⇕
−Πa,n−2(p ′′ n) = Πa,n−2(Πw,n−2f) in I
⇕
− p ′′ n = Πw,n−2f in I.
There are several reasons to prefer the new formulation (9.4.2). First of all, inequality
(9.3.9) yields
(9.4.3) p ′ nw ≤ C4 sup
≤ C4 Πw,n−2fw sup
φ∈P 0 n
φ≡0
φw
φ ′ w
φ∈P 0 n
φ≡0
Fw,n(φ)
φ ′ w
≤ C5 f L 2 w (I), ∀n ≥ 2.
To get (9.4.3), we used the Schwarz inequality, relation (6.2.7) and inequality (5.7.4).
According to the usual terminology, (9.4.3) is known as a stability condition. Namely,
a certain norm of the solution pn is bounded independently of n. In addition, pn
converges to the solution U of (9.3.4) when n tends to infinity. The proof is simple,
being byproduct of a very general result (see strang(1972)).