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260 Polynomial Approximation of Differential Equations
(11.2.26) pn,k(sk) = pn,k+1(sk) 1 ≤ k ≤ m − 1,
(11.2.27) pn,1(s0) = σ.
Another approximation is obtained by replacing condition (11.2.26) by
(11.2.28)
1
sk+1 − sk−1
(sk −sk−1)(p ′ n,k+Apn,k) + (sk+1 −sk)(p ′
n,k+1+Apn,k+1) (sk)
+ γk pn,k − pn,k+1 (sk) = f(sk) 1 ≤ k ≤ m − 1,
where γk, 1 ≤ k ≤ m, are suitable constants. We note that using (11.2.28) the global
approximating function in [s0,sm] is not continuous. Nevertheless, we still observe a
spectral rate of convergence.
Conditions at the interface points for linear or nonlinear first-order time-dependent
problems (see sections 10.3, 10.4 and 10.5) are considered for instance in patera(1984),
kopriva(1986), macaraeg and streett(1986), canuto and quarteroni(1987), fu-
naro(1990b). Other time-dependent equations are examined in pavoni(1988) and
bressan and pavoni(1990). These techniques are under development and very little is
known about the theory. Finally, the matching of equations having different orders in
the different domains is examined in gastaldi and quarteroni(1989).
11.3 Solution techniques
The matrices related to the numerical approximation of a differential problem by domain-
decomposition can be partitioned into diagonal blocks, coupled through the conditions
imposed at the interface points. The number of blocks and their size is arbitrary. Some
authors are inclined to use many subdomains and low polynomial degrees. As in the
finite element method, they achieve convergence to the exact solution by letting the