# 121c i well use the symbol kg k with the understanding

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Unformatted text preview: = 1max kf ( )k : 1 1 i N i1 Letting h = 1max jh j i N (9.1.22) i and observing that Nh (b ; a), we have je( )j h2 + (N ; 1)h3]kg( )k ~ 1 h2 1 + (b ; a)]kg( )k ] ~ 1 2 (x j; 1 x) j or je( )j C h2 where 2 (x j; 1 x) C = (1 + b ; a)kg( )k : ~ 1 8 j (9.1.23) If = x , j = 0 1 : : : N , then there is no discontinuity in the Green's function on any subinterval and the error is obtained from (9.1.19a) and (9.1.15) as j X N je(x )j j i =1 h3 kg(x )k j i j = 0 1 : : : N: i1 Following the steps leading to (9.1.23), we again nd that je(x )j C h2 j = 0 1 : : : N: j (9.1.24) Thus, the global and pointwise errors are both O(h2). This occurs because of the low polynomial degree and the arbitrary choice of the collocation points. With either higherdegree polynomials or a special choice of collocation points we can reduce the pointwise error relative to the global error. This phenomenon is called nodal superconvergence. De nition 9.1.2. Nodal superconvergence implies that the collocation solution on the mesh fa = x0 <...
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