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286 Polynomial Approximation of Differential Equations
(13.2.7)
=
= −
n
j=0
1
n
j=0
−1
1
∂pn
∂x
−1
=
n
i=0
j=0
∂pn
∂x
∂φ
∂x
+ ∂pn
∂y
∂φ
(x,η
∂x
(n)
j ) dx ˜w (n)
j
2 ∂ pn
φ (x,η
∂x2 (n)
j ) dx ˜w (n)
j
n
i=0
j=0
= −
n
i=0
j=0
∂φ
(η
∂y
(n)
i ,η (n)
j ) ˜w (n)
i ˜w (n)
j
+
−
n
i=0
n
i=0
1
−1
1
−1
∂pn
∂y
[∆pn φ](η (n)
i ,η (n)
j ) ˜w (n)
i ˜w (n)
j
∂φ
(η
∂y
(n)
i ,y) dy ˜w (n)
i
2 ∂ pn
φ (η
∂y2 (n)
i ,y) dy ˜w (n)
i
[fφ](η (n)
i ,η (n)
j ) ˜w (n)
i ˜w (n)
j =: Fn(φ), ∀φ ∈ P ⋆,0
n .
Basically, the expression above follows from (13.2.6) by replacing the integrals with
the help of a quadrature formula based on the points belonging to ℵn. Finally, we
can estimate a certain norm of the error |U − pn| with a suitable generalization
of theorem 9.4.1. The rate of convergence is obtained by examining the error due
to the use of the quadrature in place of the exact integrals. For example, the error
|F(φ) − Fn(φ)|, φ ∈ P ⋆,0
n , converges to zero with a rate depending on the smoothness
of the function f (see canuto, hussaini, quarteroni and zang(1988), section 9.7).
As usual, the analysis of the other Jacobi cases is more complicated. The Chebyshev
case (α = β = −1 2 ) is treated for instance in bressan and quarteroni(1986b). The
strategy in this class of proofs is to deal with the two variables x and y separately, in
order to take advantage of the theory developed for the one-dimensional case.
Of course, alternate spectral discretizations of Poisson’s equation, such as the tau
method, may be considered. Further generalizations result from considering approx-
imating polynomials of different degrees with respect to the variables x and y. The
same techniques apply to other partial differential equations defined in Ω. For exam-
ple, spectral methods have been successfully applied to approximate the Navier-Stokes