Untitled - Cdm.unimo.it

208 Polynomial Approximation of Differential Equations
(9.5.2) X ≡ H 1 0,v(I) := {φ |φ ∈ H 1 v(I), φ(0) = 0}.
According to section 5.7, the norm in X is given by φX :=
I φ2vdx +
∀φ ∈ X. Then, for g ∈ L2 v(I), we are concerned with finding U ∈ X such that
(9.5.3)
+∞
U
0
′ (φv) ′ +∞
dx + µ
0
Uφv dx =
+∞
0
I [φ′ ] 2 vdx
gφv dx, ∀φ ∈ X.
With the same arguments of section 9.3, using the Lax-Milgram theorem, we get exis-
tence and uniqueness of a weak solution of (9.5.3), provided the parameter µ is larger
than 1
2 max{1, 1
1−α }. The crucial part of the proof is to show coerciveness (see (9.3.6)).
This follows by virtue of lemma 8.2.7.
One checks that U ∈ X implies that limx→+∞ U(x) = 0, and the decay is exponential.
Therefore, due to theorem 6.1.4, we have the requisites to approximate the solution of
problem (9.5.3) by Laguerre functions (see section 6.7). To this end, we introduce the
subspace S 0 n := {φ| φ ∈ Sn, φ(0) = 0} ⊂ X.
For example, when g ∈ C0 ( Ī), we discretize (9.5.3) as follows. For any n ≥ 2, we seek
Pn ∈ S0 n−1 such that
+∞
(9.5.4)
0
P ′ n(φv) ′ +∞
dx + µ Pnφv dx =
0
+∞
0
( Ĩ∗ v,ng)φv dx, ∀φ ∈ S 0 n−1.
The interpolation operator Ĩ∗ v,n : C 0 ( Ī) → Sn−1, n ≥ 2, is defined by the relation
Ĩ ∗ v,ng := [ Ĩw,n(ge x )]e −x , ∀g ∈ C 0 ( Ī) (see also section 6.7), where Ĩw,n, n ≥ 1, is the
Laguerre Gauss-Radau interpolation operator (see section 3.3). We now apply theorem
9.4.1. Estimating the error g − Ĩ∗ v,ng L 2 v (I), one deduces that Pn converges to U, as n
tends to infinity, in the norm of the space X. The convergence is also uniform because
of the inequality
(9.5.5) sup
x∈I
|φ(x)e x/2 | ≤ C φX, ∀φ ∈ X.
The next step is to determine the solution to (9.5.4). We consider the substitutions:
pn(x) := Pn(x)ex , x ∈ Ī, n ≥ 2, and f(x) := g(x)ex , x ∈ Ī. Therefore, integration by
parts allows us to write (note that the boundary terms are vanishing):
1
2 ,