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Eigenvalue Analysis 179
for spectral methods, due to their global nature. On the other hand, if the boundary
conditions are well treated, results are really competitive. An example of a second-order
eigenvalue problem is examined in section 12.4.
Let us finally consider the eigenvalues associated to the system (7.4.22). These
converge to the eigenvalues of the problem
⎧
⎨φ
(8.6.9)
⎩
IV = λ φ in I =] − 1,1[,
φ(±1) = φ ′ (±1) = 0.
Here, λ assumes a countable number of positive values corresponding to the roots of
the equation cosh2 4√ λ cos2 4√ λ = 1 (see courant and hilbert(1953), Vol.1, p.296).
No asymptotic estimates can be recovered for the eigenvalues of problem (8.1.1).
In fact, the equation φ ′ = λφ in ] − 1,1[, φ(−1) = σ, has only the solution φ ≡ 0
when σ = 0.
8.7 Multigrid method
We already noticed that the speed of convergence of the Richardson method strongly
depends on the location of the eigenvalues λm, 1 ≤ m ≤ n, of the matrix D in (7.6.1).
Let ¯pm ∈ C n , 1 ≤ m ≤ n, denote the eigenvectors of D, suitably normalized. In the
computation ¯p → M ¯p, different choices of the parameter θ will result in a different
treatment of the components (modes) of the vector ¯p, along the directions selected
by ¯pm, 1 ≤ m ≤ n. For instance, let us assume that the λm’s are real, distinct,
positive and in increasing order. By examining relation (8.3.5), we deduce that the
maximum damping of the components corresponding to a large integer m (high modes),
is obtained for small values of θ. Conversely, low modes decay fast for larger values
of the parameter θ, provided the assumptions in (7.6.3) are satisfied. Therefore, for
ill-conditioned matrices we cannot expect a strong reduction of all the modes.
This can be partly overcome by using the multigrid method. We illustrate this procedure
with an example. We denote by D (n) , n ≥ 2, the (n+1)×(n+1) matrix of the system