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210 Polynomial Approximation of Differential Equations
9.6 Other techniques
Variations of the principal methods exposed in the previous sections have been sug-
gested by different authors, with the aim of preserving spectral accuracy and facilitate
numerical implementation.
For example, we already noted that discretizations by the collocation method of lin-
ear differential operators with non-constant coefficients, are much easier to implement
than those using tau method. The discrepancy is more remarkable for nonlinear equa-
tions (see sections 3.3 and 9.8). However, we can still work in the frequency space by
combining different techniques. For instance, we modify problem (9.2.6) by setting
(9.6.1)
⎧
⎨p
⎩
′ n + Πw,n−1[ Ĩw,n(Apn)] = Πw,n−1f
pn(−1) = σ.
in ] − 1,1],
We can solve the linear system corresponding to (9.6.1) by an iterative method. In this
way, whenever we need to evaluate the product Apn , we can go to the physical space
by Fourier transform (see section 4.1) and perform this operation therein. We return to
the frequency space by the inverse transform. Calculations are faster in the Chebyshev
case, where we can use the FFT (see section 4.3). We remark that this algorithm does
not coincide with a genuine tau method. The computations, however, exhibit a spectral
accuracy.
Other methods have been developed starting from variational techniques. We con-
sider for instance problem (9.1.4), which has a weak formulation given by (9.3.2)-(9.3.3)-
(9.3.4) for σ1 = σ2 = 0. The Galerkin approximation is presented in (9.4.14) which
is equivalent to (9.4.15). Assuming that w ≡ 1, from the orthogonality of Legendre
polynomials, we easily write the corresponding linear system, where the unknowns are
the Fourier coefficients of pn, n ≥ 2. Using the same weight function, a different choice
of the polynomial basis leads to the so called spectral element method introduced in
patera(1984) (see also section 11.2). In this method, pn is expanded in terms of
Chebyshev polynomials, i.e. pn = n
k=0 ckTk. Therefore, substituting in (9.4.14), one
obtains