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Domain-Decomposition Methods 259
In (11.2.18), the c (2)
j,k ’s are the Fourier coefficients of the polynomial p′′ n,k , which,
thanks to the results of section 7.1, can be expressed in terms of the cj,k’s. For any
1 ≤ k ≤ m, the quantities fj,k, j ∈ N, are the Fourier coefficients of f in ¯ Sk with
respect to uj,k, j ∈ N. In the Legendre case, (11.2.18)-(11.2.21) are equivalent to the
variational problem (11.2.11), for a suitable right-hand side function F.
We finally note that the Legendre-Galerkin method is obtained by modifying the
right-hand side of (11.2.11) by F(φ) :=
fφdx, ∀φ ∈ Xn.
I
Of course, different discretizations can be used in each subdomain. An example
of a problem defined in an unbounded domain, solved by coupling Legendre and La-
guerre approximations, is presented in section 12.2. The coupling of finite element and
spectral methods is considered in bernardi, maday and sacchi-landriani(1989) and
bernardi, debit and maday(1990).
First-order problems are approached in a similar way. For example, the differential
equation (see also (9.1.3))
(11.2.22)
can be written as
(11.2.23)
⎧
⎪⎨
⎧
⎨U
′ + AU = f in ]s0,sm],
⎩
U(s0) = σ,
U ′ k + AUk = f in ]sk−1,sk[, 1 ≤ k ≤ m − 1,
U ′ m + AUm = f in ]sm−1,sm],
Uk(sk) = Uk+1(sk) 1 ≤ k ≤ m − 1,
⎪⎩
U1(s0) = σ,
where Uk is the restriction of U to the set [sk−1,sk], 1 ≤ k ≤ m.
A multidomain collocation scheme is obtained by finding the polynomials pn,k ∈ Pn,
1 ≤ k ≤ m, such that
(11.2.24) (p ′ n,k + Apn,k)(θ (n,k)
i ) = f(θ (n,k)
i ) 1 ≤ i ≤ n − 1, 1 ≤ k ≤ m − 1,
(11.2.25) (p ′ n,m + Apn,m)(θ (n,m)
i ) = f(θ (n,m)
i ) 1 ≤ i ≤ n,