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Unformatted text preview: =1 e ( )L ( ) + R( ): (9.3.12) 0 ik k Di erentiating and using (9.3.6) yields the result (9.3.7b).
This result is not as sharp as it could be as described by the next Theorem. Theorem 9.3.2. Suppose that the collocation points are distinct with i 1< i 2< < , i = 1 2 : : : N , the one-step method (9.2.7b) is accurate to O(h ), and A(x) b(x) 2
C (a b). Then
p jY(x ) ; y(x )j = O(h )
p i i jY( ) ; y( )j = O(h +1) + O(h )
J ij ij i p i = 0 1 ::: N j = 1 2 ::: J
23 (9.3.13a) i = 0 1 : : : N: (9.3.13b) Remark. If p > 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...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14
- The Land