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Domain-Decomposition Methods 263
11.4 Overlapping multidomain methods
To avoid complex situations and cumbersome notation, we confine ourselves to the case
of two subdomains only. These are given by S1 :=]s0,s2[ and S2 :=]s1,s3[, where
s0 < s1 < s2 < s3. The set S1 ∩ S2 =]s1,s2[ = ∅ is the overlapping region. Let us
consider problem (11.2.1)(m = 3), which is equivalent to writing
(11.4.1)
⎧
⎨
−U ′′
1 = f in S1,
⎩
U1(s0) = σ1, U1(s2) = U2(s2),
⎧
⎨
−U ′′
2 = f in S2,
⎩
U2(s1) = U1(s1), U1(s3) = σ2,
where U1 and U2 are the restrictions of U to the intervals ¯ S1 and ¯ S2 respectively.
The functions U1 and U2 coincide in S1 ∩ S2.
With the intent of using the collocation method, in analogy with (11.2.3), we first
define the collocation nodes
(11.4.2) θ (n,k)
j
:= 1
(sk+1 − sk−1)η
2
(n)
j
+ sk+1 + sk−1
0 ≤ j ≤ n, 1 ≤ k ≤ 2.
For any n ≥ 2, we would now like to determine two polynomials pn,k ∈ Pn, 1 ≤ k ≤ 2,
such that
(11.4.3)
(11.4.4)
⎧
⎨
−p ′′ n,1(θ (n,1)
i ) = f(θ (n,1)
i ) 1 ≤ i ≤ n − 1,
⎩
pn,1(s0) = σ0, pn,1(s2) = pn,2(s2),
⎧
⎨
−p ′′ n,2(θ (n,2)
i ) = f(θ (n,2)
i ) 1 ≤ i ≤ n − 1,
⎩
pn,2(s1) = pn,1(s1), pn,2(s3) = σ2.
The value pn,1(s1) is obtained from the values pn,1(θ (n,1)
i ), 0 ≤ i ≤ n, by interpolation
(see formula (3.2.7)). The same argument holds for the value pn,2(s2). The first step
is to show that limn→+∞ pn,k = Uk in ¯ Sk, 1 ≤ k ≤ 2. This result is proven in
canuto and funaro(1988) in the Legendre case for any choice of the sk’s, and in
the Chebyshev case when the size of the overlapping region is sufficiently large. Note