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170 Polynomial Approximation of Differential Equations
an exact expression is given for the preconditioned eigenvalues. Namely, we have
(8.4.3) Λn,m = m(m + 1)
2 π sin 2n
sin mπ
2n
cos π
2n
sin (m+1)π
2n
, 1 ≤ m ≤ n − 1.
Thus, we obtain 1 ≤ Λn,m < π2
4 , 0 ≤ m ≤ n.
We apply the Richardson method to the system in (8.3.7) with θ := 2/(π2 /4 + 1).
This choice minimizes the spectral radius of M = I − θR −1 D, which takes the value
ρ = (π 2 /4 − 1)/(π 2 /4 + 1) ≈ 0.42 . Starting from the initial guess ¯p (0) = R −1 ¯q
(see (7.6.2)), one achieves the exact solution to the system (to machine accuracy in
double precision) in about twenty iterations. Though we do not have access to the
explicit expression of the preconditioned eigenvalues for different values of α and β,
the behavior is more or less the same.
We note that, in practice, the matrix R −1 D is not actually required. Each iteration
consists of two steps. In the first step, we evaluate the matrix-vector multiplication
¯p → D¯p. This can be carried out with the help of the FFT (see section 4.3) in the
Chebyshev case. In the second step, we compute D¯p → R −1 D¯p by solving a tridiagonal
linear system. In this way, we avoid storing the whole matrix R −1 . Moreover, the cost
of this computation is proportional to n (see isaacson and keller(1966), p.55).
Symmetric preconditioners, based on finite element discretizations, have been pro-
posed in deville and mund(1985) and in canuto and quarteroni(1985). They allow
us to apply a variation of the conjugate gradient iterative method for solving (8.3.7).
A standard trick is to accelerate the convergence by updating the parameter θ at
each iteration. This can be done in several ways. For brevity, we do not investigate this
here. A survey of the most used preconditioned iterative techniques in spectral methods
is given in canuto, hussaini, quarteroni and zang(1988), p.148. Many practical
suggestions are provided in boyd(1989).
We note that R is ill-conditioned. This introduces small rounding errors when
evaluating R −1 . Hence, after the preconditioned Richardson method has reached a
steady solution, we suggest that it be refined by performing a few iterations of the
unpreconditioned Richardson method to remove the rounding errors.