Untitled - Cdm.unimo.it

Ordinary Differential Equations 187
(9.2.13) Ĩw,nU ′ − ( Ĩw,nU) ′ w,n ≤ γ −1
1 Ĩw,nU ′ − ( Ĩw,nU) ′ L 2 w (I)
≤ γ −1
1
Ĩw,nU ′ − U ′ L 2 w (I) + ( Ĩw,nU − U) ′ L 2 w (I)
, n ≥ 1.
Finally, we use inequalities like (6.6.1) and (6.6.8) to estimate the last term. This shows
that sn L 2 w (I) tends to zero for n → +∞. Thus, we get (9.2.8).
As seen from the proof given above, the rate of convergence in (9.2.8) is the typical
one of spectral approximations. In particular, for analytic solutions U, the decay of
the error is exponential. The extension of theorem 9.2.1 to other Jacobi weights is not
straightforward. Convergence estimates for the Chebyshev case can be derived from
the results in solomonoff and turkel(1989). The use of collocation nodes relative to
Gauss type formulas is helpful in the theoretical analysis, since it allows us to substitute
summations with integrals as in (9.2.11). Unfortunately, when the weight function w
is singular at the boundary, the integrals cannot be manipulated further. This is the
major drawback in the analysis of approximations involving Chebyshev polynomials.
For example, the quantity
I φ′ φwdx, φ ∈ Pn, φ(−1) = 0 is not in general positive
when w(x) = 1/ √ 1 − x 2 , x ∈ I (see gottlieb and orszag(1977), p.89).
First, we show the results of numerical experiment when U is a very smooth func-
tion. We choose σ = 0 and A(x) := 1+x2 , x ∈ Ī. The datum f in (9.1.3) is such that
U(x) := sin (1 + x), x ∈ Ī. In table 9.2.1, we give the error En := pn − Ĩw,nU L 2 w (I) for
various n, when pn is obtained with the scheme (9.2.7) relative to the Legendre nodes.
It is evident that the convergence is extremely fast. Note that, from (9.2.9), the error
pn − U L 2 w (I) is bounded by En plus a term which does not depend on the poly-
nomial pn. Therefore, the examination of En gives sufficient information about the
efficiency of the approximation technique. In addition, according to formula (3.8.10),
the computation of En is obtained from the values of pn at the nodes. These values
are the actual solutions of (9.2.7), after inverting the linear system corresponding to
(7.4.7).