48 or 749a and calculate an error estimate by solving

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Unformatted text preview: o yield Ae(V E ) = (V f )e ; Ae(V U ) ~ 8V 2 S N : (7.4.17a) 26 Analysis of the Finite Element Method or Ae(V E ) = (V r)e ~ 8V 2 S N : (7.4.17b) Yu 22, 23] used these arguments to prove asymptotic convergence of error estimates to true errors for elliptic problems. Adjerid et al. 2, 3] obtained similar results for transient parabolic systems. Proofs, in both cases, apply to a square region with square p elements of spacing h = 1= N . A typical result follows. 1 N Theorem 7.4.1. Let u 2 HE \ H p+2 and U 2 SE be solutions of (7.4.2) using complete piecewise-bi-polynomial functions of order p. 1. If p is an odd positive integer then ke( )k2 = kE ( )k2 + O(h2p+1 ) 1 1 (7.4.18a) N24 h2 X X X U (P )]2 kE k = 16(2p + 1) e=1 i=1 k=1 xi k e i (7.4.18b) where 2 1 Pk e, k = 1 2 3 4, are the coordinates of the vertices of e, and f (P)]i denotes the jump in f (x) in the direction xi , i = 1 2, at the point P. 2. If p is a positive even integer then (7.4.18a) is satis ed with Ae(Vi E ) = (V f )e ; Ae(Vi U ) (7.4.18c) where E (x1 x2 ) = b1 e Vi(x1 x2 ) =...
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