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Unformatted text preview: o square elements is shown on the
left of Figure 8.3.1. The seconddegree approximation (shown at the top left) consists of a
bilinear shape function at each vertex and a seconddegree correction on each edge. The
fourthdegree approximation (bottom left) consists of bilinear shape functions at each
vertex, second, third and fourthdegree corrections on each edge, and a fourthdegree
bubble function associated with the centroid (cf. Section 4.4). The maximum degree of
the polynomial associated with a mesh entity is identi ed on the gure. The second and
fourthdegree shape functions would be incompatible (discontinuous) across the common
edge between the two elements. This situation can be corrected by constraining the
edge functions to the lowerdegree (two) basis of the top element as shown in the center 8.3. p and hpRe nement 15 Figure 8.2.15: Solution of Example 8.2.1 by uniform ( ) and adaptive ( ) hre nement
33].
portion of the gure or by adding third and fourthorder edge functions to the upper
element as shown on the right of the gure. Of the two possibilities, the addition of the
higher degree functions is the most popular. Constraining the space to the lowerdegree
polynomial could result in a situation where error criteria satis ed on the element on the
lower left of Figure 8.3.1 would no longer be satis ed on the element in the lowercenter
portion of the gure.
Remark 1. The incompatibility problem just described would not arise with the
hierarchical data structures de ned in Section 5.3 since edge functions are blended onto
all elements containing the edge and, hence, would always be continuous.
Szabo 39] developed a strategy for the adaptive variation of p by constructing error
estimates of solutions with local degrees p, p ; 1, and p ; 2 on Element e and extrapolating
to get an error estimates for solutions of higher degrees. With a hierarchical basis, this
is straightforward when p > 2. One could just use the di erences between higher and
lowerorder solutions or an error estimation procedure as described in Section 7.4. When
p = 2 on Element e, local error estimates of solutions having degrees 2 and 1 are linearly
extrapolated. Szabo 39] began by generating piecewiselinear (p = 1) and piecewisequadratic (p = 2) solutions everywhere and extrapolating the error estimates. Flaherty
and Moore 20] suggest an alternative when p = 1. They obtain a \lowerorder" piecewise 16 Adaptive Finite Element Techniques
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01 Figure 8.3.1: Second and fourthdegree hierarchical shape functions on two square elements are incompatible across the common...
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 Spring '14
 JosephE.Flaherty

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