# 418b where 2 1 pk e k 1 2 3 4 are the coordinates of

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Unformatted text preview: xi and m (x) e p+1 (x ) + b p+1 1 2 e e (x2 ) e p+1 (x1 ) p+1 (x2 ) e e x1 x2 i=1 2 (7.4.18d) (7.4.18e) is the mapping of the hierarchical basis function r ) = 2m2; 1 3 from ;1 1] to the appropriate edge of e . N m( Z ;1 Pm;1 ( )d (7.4.18f) Proof. cf. Adjerid et al. 2, 3] and Yu 22, 23]. Coordinates are written as x = x1 x2 ]T instead of (x y) to simplify notation within summations. The hierarchical basis element (7.4.18f) is consistent with prior usage. Thus, the subscript 3 refers to a midside node as indicated in Figure 7.4.2. 7.4. A Posteriori Error Estimation 27 Remark 1. The error estimate for even-degree approximations has di erent trial and test spaces. The functions Vi(x1 x2) vanish on @ e . Each function is the product of a \bubble function" p+1(x1 ) p+1(x2 ) biased by a variation in either the x1 or the x2 e e direction. As an example, consider the test functions on the canonical element with p = 2. Restricting (7.4.18e) to the canonical element ;1 1 2 1, we have 3 3 Vi(...
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