Untitled - Cdm.unimo.it

Domain-Decomposition Methods 253
Theorem 11.2.1 - Let Uk, 1 ≤ k ≤ m, be the solutions of problem (11.2.2) with
σ1 = σ2 = 0. Let η (n)
j , 0 ≤ j ≤ n, in (11.2.3), be the nodes of the Legendre (w ≡ 1)
Gauss-Lobatto formula (3.5.1). Let pn,k, 1 ≤ k ≤ m, be the polynomials satisfying
(11.2.4), (11.2.5), (11.2.7) with σ1 = σ2 = 0, and (11.2.8) with γk := n(n+1)
sk+1−sk−1 ,
1 ≤ k ≤ m − 1. Then, we have
(11.2.9) lim
n→+∞
sk
sk−1
(Uk − pn,k) 2 dx +
sk
sk−1
Proof - For any 1 ≤ k ≤ m, define the weights ˜w (n,k)
j
(U ′ k − p ′ n,k) 2 dx
1
2
= 0, 1 ≤ k ≤ m.
:= 1
2 (sk −sk−1) ˜w (n)
j , 0 ≤ j ≤ n
(see (3.5.6)). Therefore, by (3.5.1) with w ≡ 1, we get the integration formula
(11.2.10)
sk
sk−1
p dx =
n
j=0
p(θ (n,k)
j ) ˜w (n,k)
j
∀p ∈ P2n−1, 1 ≤ k ≤ m.
Let Xn denote the space of continuous functions φ in I, such that φ(s0) = φ(sm) = 0
and such that the restriction of φ to any interval ¯ Sk is a polynomial of degree n. Then,
by noting that γk[ ˜w (n,k)
n
satisfies
(11.2.11)
= −
= −
m
n
k=1
i=1
m
k=1
= −
Sk
m
k=1
+ ˜w (n,k+1)
0 ] = 1, 1 ≤ k ≤ m − 1, the function πn ∈ Xn
I
π ′′
nφ dx +
sk
sk−1
[p ′′ n,kφ](θ (n,k)
i ) ˜w (n,k)
i
=
m
n
k=1
i=1
π ′ nφ ′ dx =
m−1
k=1
p ′′ n,kφ dx +
+
m−1
k=1
[fφ](θ (n,k)
i ) ˜w (n,k)
i
m
k=1
lim
x→s −
k
m−1
k=1
Sk
π ′ nφ ′ dx
(π ′ nφ)(x) − lim
x→s +
k
(π ′
nφ)(x)
p ′ n,kφ − p ′
n,k+1φ (sk)
γk( ˜w (n,k)
n
+ ˜w (n,k+1)
0 ) p ′ n,kφ − p ′
n,k+1φ (sk)
=: Fn(φ), ∀φ ∈ Xn.