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204 Polynomial Approximation of Differential Equations
To develop the convergence analysis, an extension of theorem 9.4.1 is needed, since
the bilinear form now depends on n. For the sake of simplicity, we do not discuss
this generalization. The reader finds indications to proceed in this investigation in
strang(1972) and ciarlet(1978).
A different treatment of the boundary conditions can be taken into consideration.
First, we examine the Legendre case (w ≡ 1). In (9.4.16) define Bw,n : Pn × Pn → R
and Fw,n : Pn → R such that
(9.4.21) Bw,n(ψ,φ) :=
(9.4.22) Fw,n(φ) :=
n
j=0
n
ψ
I
′ φ ′ dx + µ
j=0
ψ(η (n)
j )φ(η (n)
j ) ˜w (n)
j , ∀ψ,φ ∈ Pn,
f(η (n)
j )φ(η (n)
j ) ˜w (n)
j + σ2φ(1) − σ1φ(−1), ∀φ ∈ Pn.
We replaced the integrals in (9.3.15) and (9.3.16) by the summations. The modification
does not alter the scheme at the nodes inside the interval I. In fact, using the test
functions φ := ˜l (n)
i , 1 ≤ i ≤ n − 1, we still get the set of equations −p ′′ n(η (n)
i ) +
µpn(η (n)
i ) = f(η (n)
i ), 1 ≤ i ≤ n − 1. At the boundary points one gets
(9.4.23)
⎧
⎨ −p ′′ n(−1) + µpn(−1) − γp ′ n(−1) = f(−1) − γσ1,
⎩
−p ′′ n(1) + µpn(1) + γp ′ n(1) = f(1) + γσ2,
where γ := [ ˜w (n)
0 ] −1 = [ ˜w (n)
n ] −1 = 1
2
n(n + 1). Thus, we obtained the system (7.4.16)
with q := Ĩw,nf. Note that in (9.4.20) the equation is also collocated at the points
x = ±1, and the boundary conditions are used as corrective terms.
We can make an interesting remark. In the previous examples, the variational
formulations have been presented as an intermediate step to obtain convergence results
by virtue of general abstract theorems. Here, we followed an inverse path. Starting
from an appropriate weak formulation, we recovered a new approximation scheme with
different conditions at the boundary. These relations may not have a direct physical
interpretation. Nevertheless, convergence is still achieved. As a matter of fact, one
shows that limn→+∞ p ′ n(−1) = σ1, limn→+∞ p ′ n(1) = σ2 (see funaro(1988)).