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Unformatted text preview: btaining the solution. With a hierarchical embedding, computations needed for the lower-order method
are also needed for the higher-order method and, hence, need not be repeated.
The extrapolation techniques just described are typically too expensive for use as
error estimates. We'll develop a residual-based error estimation procedure that follows
Bank (cf. 8], Chapter 7) and uses many of the ideas found in order embedding. We'll
follow our usual course of presenting results for the model problem
;r pru + qu = ;(pux )x ; (puy )y + qu = f (x y ) u(x y) = (x y) 2 @ pun(x y) = E (x y) 2
(x y) 2 @ (7.4.1a)
N (7.4.1b) however, results apply more generally. Of course, the Galerkin form of (7.4.1) is: deter1
mine u 2 HE such that A(v u) = (v f )+ < v >
(v f ) = A(v u) = ZZ ZZ 1
8v 2 H0 (7.4.2a) vfdxdy (7.4.2b) prv ru + qvu]dxdy (7.4.2c) and < v u >= Z N
Similarly, the nite element solution U 2 SE @N vuds: (7.4.2d) 1
HE satis es A(V U ) = (V f )+ < V > N
8V 2 S0 : (7.4.3) 20 Analysis of the Finite Element Method We...
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- Spring '14