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76 Polynomial Approximation of Differential Equations
4.4 Other fast methods
There are situations in which a polynomial, defined by the values assumed at the nodes
of some Gauss integration formula, must be determined at the points corresponding to
the relative Gauss-Lobatto formula. More precisely, from (3.2.2), we get for p ∈ Pn−1
n
(4.4.1) p(η (n)
i ) =
j=1
Conversely, if p ∈ Pn, we obtain by (3.2.7)
n
(4.4.2) p(ξ (n)
i ) =
j=0
p(ξ (n)
j ) l (n)
j (η (n)
i ), 0 ≤ i ≤ n.
p(η (n)
j ) ˜ l (n)
j (ξ (n)
i ), 1 ≤ i ≤ n.
As usual, particularly interesting is the Chebyshev case, where this interpolation process
can be optimized. By (4.1.4) or (4.1.8)-(4.1.9), we recover the Fourier coefficients using
the FFT. Then, in place of (4.4.1), one computes
(4.4.3) p(η (n)
i ) =
n−1
j=0
cj Tj(η (n)
n−1
i ) = −
Furthermore, in place of (4.4.2), we can write
n
(4.4.4) p(ξ (n)
i ) =
j=0
cj Tj(ξ (n)
i ) = −
so that an FFT-like algorithm can still be used.
n
j=0
j=0
cj cos
cj cos ij
π, 0 ≤ i ≤ n.
n
j(2i − 1)
π, 1 ≤ i ≤ n,
2n
The advantages of using two sets of points, like those considered above, in the dis-
cretization of certain classes of partial differential equations, such as the incompressible
Navier-Stokes equations (see section 13.4), have been investigated in malik, zang and
hussaini(1985), and theoretically analyzed and extended in bernardi and maday
(1988). The need for efficient transfer of data between the two grids, could favor the
Chebyshev polynomials for these kind of computations.
Additional algorithms, based on the cosine FFT described in the previous section,
have been adapted for other Jacobi polynomials (see orszag(1986)). Unfortunately,
they are deduced from suitable asymptotic expansions. Therefore, they are fast and
accurate only when n is quite large.