Untitled - Cdm.unimo.it

264 Polynomial Approximation of Differential Equations
that, for any n ≥ 2, the polynomials pn,1 and pn,2 do not coincide in S1 ∩ S2. The
construction of the linear system relative to (11.4.3)-(11.4.4), and the generalization to
the case of more domains, is left to the reader.
Finally, as in the previous section, we suggest an iterative algorithm for the solution
of the linear system corresponding to (11.4.3)-(11.4.4). For any y ∈ R, we construct
two polynomials qn,k ∈ Pn, 1 ≤ k ≤ 2, satisfying
(11.4.5)
(11.4.6)
⎧
⎨ −q ′′
n,1(θ (n,1)
i ) = f(θ (n,1)
i ) 1 ≤ i ≤ n − 1,
⎩
qn,1(s0) = σ0, qn,1(s2) = y,
⎧
⎨ −q ′′
n,2(θ (n,2)
i ) = f(θ (n,2)
i ) 1 ≤ i ≤ n − 1,
⎩
qn,2(s1) = qn,1(s1), qn,2(s3) = σ2.
Thus, we define the mapping Γ : R → R such that Γ(y) := qn,2(s2). Therefore,
qn,k ≡ pn,k in ¯ Sk, 1 ≤ k ≤ 2, if and only if Γ(y) = y. The aim is now to find
the fixed point y ∗ of the function Γ. This can be determined by the recursion formula
ym+1 = Γ(ym), m ∈ N, where y0 is given. At any step, we have to solve the collocation
problems (11.4.5) and (11.4.6). Such a procedure is known as the Schwarz alternating
method (see schwarz(1890), Vol.2). In canuto and funaro(1988), the authors prove
that limn→+∞ ym = y ∗ , for Legendre and Chebyshev approximations. In particular, in
the Legendre case, we get the relation
(11.4.7) |ym − y ∗ | ≤ κm
1 − κ |y1 − y0| m ∈ N, where κ := (s1 − s0)(s3 − s2)
(s2 − s0)(s3 − s1) .
Note that the parameter κ < 1 approaches 1 when the size of the overlapping region
tends to zero.
A presentation of the Schwarz alternating method in a very general framework is
given in dryja and widlund(1990), where a unifying theory provides a link between
overlapping and non-overlapping multidomain methods.