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288 Polynomial Approximation of Differential Equations
where Ân is the (n − 1) × (n − 1) matrix { ˜ d (2)
ij } 1≤i≤n−1
1≤j≤n−1
(see (13.2.4) for n = 4).
In section 8.2 we claimed that for −1 < α < 1 and −1 < β < 1, the eigenvalues of
− Ân are real positive and distinct, although this cannot always be proven.
Under this assumption, we can find an invertible (n − 1) × (n − 1) matrix Bn such
that Ân = B −1
n ΛnBn, where Λn is diagonal. At this point, one easily checks that
(Bn ⊗ Bn) −1 D ⋆ n(Bn ⊗ Bn) is equal to the diagonal matrix În ⊗ Λn + Λn ⊗ În. This
shows that the eigenvalues of −D ⋆ n can be recovered from those of problem (8.2.1). In
particular, they are λn,m + λn,k, 1 ≤ m ≤ n − 1, 1 ≤ k ≤ n − 1. From (13.3.2), we
finally obtain the following relation (see lynch, rice and thomas(1964)):
(13.3.4) (D ⋆ n) −1 = (Bn ⊗ Bn)( În ⊗ Λn + Λn ⊗ În) −1 (B −1
n ⊗ B −1
n ).
This suggests the possibility of performing the multiplication ¯ fn → (D ⋆ n) −1 ¯ fn very
efficiently, provided we know how to decompose the block Ân.
System (13.2.3) can be also solved by a preconditioned iterative method (see section
8.3). Following section 8.4, we can construct a (n−1)×(n−1) preconditioning matrix
ˆRn for − Ân, based on a finite-difference scheme at the collocation nodes. Then, one
finds that the (n − 1) 2 × (n − 1) 2 matrix Rn := În ⊗ ˆ Rn + ˆ Rn ⊗ În is a good
preconditioner for −D ⋆ n, since the eigenvalues of R −1
n D ⋆ n and ˆ R −1
n Ân have the same
qualitative behavior. In addition, since Rn has the same structure of D ⋆ n, we can
invert it by the procedure described above.
Another iterative technique is the alternating-direction method, which is detailed
in peaceman and rachford(1955), douglas and rachford(1956), varga(1962).
Let us fix the the real parameter ω > 0 and denote by In the (n − 1) 2 × (n − 1) 2
identity matrix. The algorithm consists of two successive steps, i.e.,
(13.3.5) (−An + ωIn) ¯q (k)
n := ¯ fn + (KnAnKn + ωIn) ¯p (k)
n , k ∈ N,
(13.3.6) (−KnAnKn + ωIn) ¯p (k+1)
n
where the initial guess ¯p (0)
n
:= ¯ fn + (An + ωIn) ¯q (k)
n , k ∈ N,
is assigned and {¯q (k)
n }k∈N is an auxiliary sequence of
vectors. Then we have limk→+∞ ¯p (k)
n = ¯pn, where ¯pn is the solution of (13.2.3).