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252 Polynomial Approximation of Differential Equations
(11.2.5) pn,k(sk) = pn,k+1(sk) 1 ≤ k ≤ m − 1,
(11.2.6) p ′ n,k(sk) = p ′ n,k+1(sk) 1 ≤ k ≤ m − 1,
(11.2.7) pn,1(s0) = σ1, pn,m(sm) = σ2.
Basically, in (11.2.4) we collocated the differential equation at the nodes inside Sk,
1 ≤ k ≤ m, and imposed the boundary conditions (11.2.7). Relations (11.2.5) and
(11.2.6) require the continuity of πn and its derivative in the interval I, i.e., πn ∈ C 1 (I).
It is clear that (11.2.4)-(11.2.7) is equivalent to a m(n+1)×m(n+1) linear system. To
determine the corresponding matrix, we argue as in sections 7.2 and 7.4. For instance,
the entries of the derivative matrix at the collocation nodes θ (n,k)
j , 0 ≤ j ≤ n, are given
by 2 ˜ d (1)
ij /(sk − sk−1) 0≤i≤n
0≤j≤n
, 1 ≤ k ≤ m.
In the ultraspherical case, we can replace (11.2.6) by
(11.2.8)
−1
sk+1 − sk−1
+ γk
(sk − sk−1)p ′′ n,k + (sk+1 − sk)p ′′
n,k+1 (sk)
p ′ n,k − p ′
n,k+1 (sk) = f(sk) 1 ≤ k ≤ m − 1,
where γk, 1 ≤ k ≤ m, are suitable constants. In this case we only have πn ∈ C 0 (I).
The use of formula (11.2.8) was proposed in funaro(1986) for Legendre nodes and
funaro(1988) for Chebyshev nodes. This new condition, which is closely related to
(9.4.23), results from a variational formulation of problem (11.2.1). When the constants
γk in (11.2.8) are chosen appropriately, numerical results are in general more accurate
than those obtained with condition (11.2.6). In both cases, we expect convergence of
the approximating function πn to the exact solution U when n → +∞. A convergence
analysis can be developed following the variational approach described in sections 9.3
and 9.4. To illustrate the basic principles, we show the theory in a simple case. We
recall that there is no loss of generality if we assume σ1 = σ2 = 0 in (11.2.1).