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Domain-Decomposition Methods 251
Consider first the second-order boundary-value problem (see also (9.1.4)):
(11.2.1)
⎧
⎨ −U ′′ = f in I,
⎩
U(s0) = σ1, U(sm) = σ2,
where σ1,σ2 ∈ R, and f : I → R is a given continuous function.
Given the decomposition of the domain I, (11.2.1) is equivalent to finding m functions
Uk : ¯ Sk → R, 1 ≤ k ≤ m, such that
(11.2.2)
⎧
−U ′′
k = f in Sk, 1 ≤ k ≤ m,
⎪⎨ Uk(sk) = Uk+1(sk) 1 ≤ k ≤ m − 1,
U ′ k (sk) = U ′ k+1 (sk) 1 ≤ k ≤ m − 1,
⎪⎩
U1(s0) = σ1, Um(sm) = σ2.
Of course, Uk turns out to be the restriction of U to the set ¯ Sk, 1 ≤ k ≤ m.
Instead of approximating the solution of (11.2.1) by a unique polynomial in the whole
interval I as suggested in section 9.4, we look for a continuous approximating function
πn : Ī → R, such that pn,k(x) := πn(x), x ∈ ¯ Sk, is a polynomial in Pn converging
for n → +∞ to the corresponding function Uk, 1 ≤ k ≤ m.
Let us examine for instance the collocation method. First, we map the points
η (n)
j ∈ [−1,1], 0 ≤ j ≤ n, in each interval ¯ Sk, 1 ≤ k ≤ m. This is done by defining the
new set of nodes
(11.2.3) θ (n,k)
j
We note that θ (n,k)
n
:= 1
2 [(sk − sk−1)η (n)
j + sk + sk−1] 0 ≤ j ≤ n, 1 ≤ k ≤ m.
= θ (n,k+1)
0
= sk, 1 ≤ k ≤ m − 1.
Then, following orszag(1980), we define the polynomials pn,k, 1 ≤ k ≤ m, to be
solutions of the set of equations
(11.2.4) −p ′′ n,k(θ (n,k)
i ) = f(θ (n,k)
i ) 1 ≤ i ≤ n − 1, 1 ≤ k ≤ m,