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Domain-Decomposition Methods 257
(11.2.14) p ′ nk,k(sk−1) − p ′ nk+1,k+1(sk+1) =
which is suggested by the identity
(11.2.15) U ′ (sk−1) − U ′ sk+1
(sk+1) = −
sk−1
sk+1
sk−1
U ′′ dx =
f dx 1 ≤ k ≤ m − 1,
sk+1
sk−1
f dx 1 ≤ k ≤ m − 1.
The integral in (11.2.14) can be computed by quadrature, for instance by the Clenshaw-
Curtis formula once the values of f at the nodes are known (see section 3.7). This
approach was suggested in macaraeg and streett(1986) and produces results com-
parable to those based on relation (11.2.6). Going back to the example previously
illustrated, (11.2.14) with m = 2 is obtained by setting in (11.2.13): τ1 := 2 ˜ d (1)
00 ,
τ2 := 2 ˜ d (1)
01 , τ3 := 2 ˜ d (1)
02 , τ4 := −2 ˜ d (1)
20 , τ5 := −2 ˜ d (1)
21 , τ6 := −2 ˜ d (1)
22 , and by replacing
the fourth entry of the right-hand side vector by 1
−1 fdx.
The spectral element method (see patera(1984) and section 9.6) uses in the Cheby-
shev case a variational formulation similar to that in (11.2.11) (we recall that this cor-
responds to an approximation of problem (11.2.1) with σ1 = σ2 = 0). In any interval
¯Sk, 1 ≤ k ≤ m, we consider the expansion pn,k = n
j=0
(11.2.16) ˜ l (n,k)
j
pn,k(θ (n,k)
j
) ˜l (n,k)
j , where
(x) := ˜l (n)
j (2x − sk − sk−1)/(sk − sk−1) , ∀x ∈ ¯ Sk, 0 ≤ j ≤ n.
The nodes are obtained from (11.2.3), where η (n)
j , 0 ≤ j ≤ n, are the Gauss-Lobatto
Chebyshev points (see (3.1.11)). The associated Lagrange polynomials ˜ l (n)
j , 0 ≤ j ≤ n,
are defined in (3.2.10). Next, we modify the right-hand side of (11.2.11) by defining
(11.2.17)
m
n
Fn(φ) := f(θ (n,k)
i ) ˜(n,k) l i φ dx ,
Sk
∀φ ∈ Xn.
k=1
i=1
The set of equations is obtained by testing on the functions φ ≡ φk,j ∈ Xn, such that
⎧
⎨˜(n,k)
l j (x)
φk,j(x) :=
⎩
if x ∈ ¯ 0
Sk
if x ∈ Ī − ¯ Sk
1 ≤ j ≤ n − 1, 1 ≤ k ≤ m,