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258 Polynomial Approximation of Differential Equations
and on the set of functions φ ≡ φk ∈ Xn, such that
⎧
˜ l (n,k)
n (x) if x ∈ ¯ Sk
⎪⎨
φk(x) :=
⎪⎩
0 elsewhere
˜ l (n,k+1)
0 (x) if x ∈ ¯ Sk+1
1 ≤ k ≤ m − 1.
The cumbersome construction of the corresponding (nm−1)×(nm−1) linear system is
left to the reader. Since we are using Chebyshev nodes instead of Legendre nodes, we can
apply the FFT (see section 4.3) in our computations. Besides, the proof of convergence
follows from theorem 9.4.1 and by showing that the term
I fφdx − Fn(φ) , where
φ ∈ Xn, tends to zero. Different polynomial degrees may be also considered in each
domain.
We can use the same techniques introduced above when the approximating poly-
nomials are represented in the frequency space. For example, the multidomain tau
method is obtained by seeking pn,k = n j=0 cj,kuj,k, 1 ≤ k ≤ m, where uj,k(x) :=
(2x − sk − sk−1)/(sk − sk−1) , x ∈ ¯ Sk, 1 ≤ k ≤ m, j ∈ N, and the uj’s are the
uj
classical orthogonal polynomials in [−1,1]. Then, in analogy with (11.2.4)-(11.2.7), we
require that
(11.2.18) − c (2)
j,k
(11.2.19)
(11.2.20)
(11.2.21)
n
j=0
n
j=0
cj,k uj,k(sk) =
cj,k u ′ j,k(sk) =
n
j=0
= fj,k
1
n
j=0
n
j=0
cj,1 uj,1(s0) = σ1,
≤ k ≤ m,
cj,k+1 uj,k+1(sk) 1 ≤ k ≤ m − 1,
cj,k+1 u ′ j,k+1(sk) 1 ≤ k ≤ m − 1,
n
cj,m uj,m(sm) = σ2.
j=0