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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 evendegree 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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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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