Chapter 9. Collocation Methods. 9.1 Introduction. Let's continue the discussion of
collocation methods with the second-order linear BVP. Ly = y. 00. + p(x)y. 0.
Chapter 9 Collocation Methods 9.1 Introduction Let's continue the discussion of collocation methods with the second-order linear BVP
Ly = y + p(x)y + q(x)y = r(x) 00
a J convergence at the nodes and collocation points is at a higher rate than implied by Theorem 9.3.1. This is the phenomenon of superconvergence that we illustrated in Section 9.1. Proof. As in Section 9.1, introduce the Green's function
e( ) =
Z
b
a
XZ
G( x)Le(x)dx =
J
j
=1
xj
xj
G( x)Le(x)dx:
;1
(9.3.14)
Let us assume that 2= (x 1 x ) and, following the logic introduced in Section 9.1, write j;
j
Y J
G( x)Le(x) = w(x) (x ; ) j
ij
=1
The function w(x) involves the J th derivative of Le. Using this and (9.3.7b), we may expect w(x) to have p ; J bounded derivatives. Thus, expand w(x) in a Taylor's series of the form w(x) = P 1(x) + O(h ) p;J
p;J ;
i
where P 1(x) is a polynomial of degree p ; J ; 1. If the one-step method is accurate to order p then p;J ;
Z
xj
xj
Since
;1
P
p; J ;
J
j
=1
(x ; )dx = 0: ij
Y J
j
we have
1(x)
Y
Z
xj
xj
;1
=1
(x ; ) = O(h ) J
ij
G( x)Le(x)dx = O(h
P ;J i
i
)O(h )h J i
i
2= (x
1
j;
x ): j
(9.3.15a)
The result (9.3.13a) is obtained by summing the above relation over the subintervals. When 2 (x 1 x ) then we are only able to show that
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