Untitled - Cdm.unimo.it

Examples 273
n En
4 2.88462
8 .164540
12 .087322
16 .049646
20 .052595
Table 12.2.1 - Errors for the approximation
of problem (12.2.1) by the scheme (12.2.2).
Another approach is to use Laguerre functions (see section 6.7) for obtaining a
global approximation of U in the whole domain I (see section 9.5). It is recommended
not to use high degree Laguerre polynomials, due to the effect of rounding errors in
the computations. Thus, we adopt a non-overlapping multidomain method (see section
11.2). For any n ≥ 2 and m ≥ 1, we approximate the solution of (12.2.1) in S1 :=]0,2[
by a polynomial pn ∈ Pn , and in S2 :=]2,+∞[ by a Laguerre function Qm ∈ Sm−1.
At the interface point x = 2 we assume the continuity of the approximating function
and its derivative. Then, we define the nodes ˜η (m)
j := η (m)
j + 2, 0 ≤ j ≤ m − 1, where
η (m)
0 := 0 and η (m)
d
j , 1 ≤ j ≤ m−1, are the zeroes of dxL(α) m for α = 0. The following
collocation scheme is considered:
⎧
(12.2.3)
−p ′′ n(ˆη (n)
i ) − p ′ n(ˆη (n)
i ) = exp(−ˆη (n)
i ) f(ˆη (n)
i ), 1 ≤ i ≤ n − 1,
⎪⎨
pn(0) = 0,
⎪⎩
pn(2) = Qm(2), p ′ n(2) = Q ′ m(2),
−Q ′′ m(˜η (m)
i ) − Q ′ m(˜η (m)
i ) = exp(−˜η (m)
i ) f(˜η (m)
i ), 1 ≤ i ≤ m − 1.
The substitution qm(x) := Qm(x)e x−2 , ∀x ∈ ¯ S2, leads to qm ∈ Pm−1, and (12.2.3)
becomes
(12.2.4)
⎧
−p ′′ n(ˆη (n)
i ) − p ′ n(ˆη (n)
i ) = exp(−ˆη (n)
i ) f(ˆη (n)
i ), 1 ≤ i ≤ n − 1,
⎪⎨
pn(0) = 0,
⎪⎩
pn(2) = qm(2), p ′ n(2) = q ′ m(2) − qm(2),
−q ′′ m(˜η (m)
i ) + q ′ m(˜η (m)
i ) = e −2 f(˜η (m)
i ), 1 ≤ i ≤ m − 1.