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Domain-Decomposition Methods 261
size of these domains tend to zero. This is the case of the h-p-version of the finite
element method (see babu˘ska, szabo and katz(1981), babu˘ska and suri(1987)). In
contrast, the common practice in spectral methods is to use large domains and high
polynomial degrees. Whether the first approach is better than the latter depends on
the circumstances.
The use of domain-decomposition methods gives us the possibility of developing
useful iterative algorithms by taking advantage of the structure of the matrices. At each
iteration, the linear equations in the system are decoupled by blocks, which are solved
independently. Of course, such a feature is particularly well adapted to computers with
a parallel architecture. We illustrate the basic ideas with an example. We study the
approximation πn of problem (11.2.1), given by the multidomain collocation scheme
(11.2.4)-(11.2.7). We first introduce some notations. Let yk, 0 ≤ k ≤ m, be a set of
real values with y0 := σ1 and ym := σ2. For any 1 ≤ k ≤ m, we define the polynomial
qn,k ∈ Pn satisfying the set of equations
(11.3.1)
⎧
⎨ −q ′′
n,k (θ(n,k) i ) = f(θ (n,k)
i ) 1 ≤ i ≤ n − 1,
⎩
qn,k(sk−1) = yk−1, qn,k(sk) = yk.
Next, we define the mapping Γ : R m−1 → R m−1 , Γ ≡ (Γ1, · · · ,Γm−1), whose compo-
nents are given by
(11.3.2) Γk(y1, · · · ,ym−1) := q ′ n,k(sk) − q ′ n,k+1(sk), 1 ≤ k ≤ m − 1.
It is clear that πn is characterized by the relation
(11.3.3) Γk(πn(s1), · · · ,πn(sm−1)) = 0, 1 ≤ k ≤ m − 1.
From the values πn(sk), 1 ≤ k ≤ m − 1, it is easy to recover the entire function πn
by noting that qn,k in (11.3.1) coincides with pn,k when yk−1 = πn(sk−1) and
yk = πn(sk). The problem is consequently reduced to finding the m − 1 zeroes of
Γ. This can be done by an iterative procedure. Using an explicit method, at each
step we solve in parallel m − 1 linear systems of dimension (n + 1) × (n + 1) corre-
sponding to (11.3.1). This suggests to constructing once for all, for any 1 ≤ k ≤ m,