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212 Polynomial Approximation of Differential Equations
When ǫ tends to zero, Uǫ converges to the discontinuous function: U0 ≡ 0 in [−1,1[,
U0(1) = 1. For very small values of ǫ, various physical phenomena are related to the
function Uǫ, which displays sharp derivatives in a neighborhood of the point x = 1. Due
to this behavior, approximations of Uǫ by low degree global polynomials, are affected
by oscillations. Nevertheless, we are not in the situation that characterizes the Gibbs
phenomenon (see sections 6.2 and 6.8, as well gottlieb and orszag(1977), p.40). Ac-
tually, since the solution is analytic for ǫ > 0, uniform convergence with an exponential
rate is obtained with the usual techniques. Tau, Galerkin and collocation methods have
been studied in canuto(1988). In particular, the tau method is equivalent to finding
pǫ,n ∈ Pn, n ≥ 2, ǫ > 0, satisfying problem (7.3.5), where: A ≡ 0, B ≡ 1
ǫ , q ≡ 0,
σ1 = 0, σ2 = 1. Thus, with the notations of section 7.1, the corresponding linear system
is
(9.7.3)
⎧
⎨
⎩
−ǫ c (2)
k + ck = 0 0 ≤ k ≤ n − 2,
n
k=0 ckuk(−1) = 0, n
k=0 ckuk(1) = 1.
The coefficients c (2)
k , 0 ≤ k ≤ n, are given for ν := α = β in karageorgis and
phillips(1989) (see also (7.1.9) and (7.1.10)). These are all positive for ν > −1.
Therefore, it is easy to deduce that ck > 0, 0 ≤ k ≤ n. This property is used in
canuto(1988) to estimate the maximum norm of pǫ,n. For instance, from (1.3.9), we
have
(9.7.4) |pǫ,n(x)| =
≤
n
k=0
ck
n + ν
n
=
n
ckuk(x)
≤
k=0
n
ck|uk(x)|
k=0
n
ckuk(1) = pǫ,n(1) = 1, ∀x ∈ Ī, ∀ǫ > 0, ∀n ≥ 2.
k=0
In addition, using relation (1.4.8), in the Legendre case one gets
(9.7.5) |pǫ,n(x)| ≤
n
k=0
ck|Pk(x)| ≤ 1
2 (1 + x2 )
n
k=0
ckPk(1) = 1
2 (1 + x2 ),
∀x ∈ Ī, ∀ǫ > 0, ∀n ≥ 2.