A situation where second and fourth degree

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Unformatted text preview: o square elements is shown on the left of Figure 8.3.1. The second-degree approximation (shown at the top left) consists of a bilinear shape function at each vertex and a second-degree correction on each edge. The fourth-degree approximation (bottom left) consists of bilinear shape functions at each vertex, second, third and fourth-degree corrections on each edge, and a fourth-degree 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 fourth-degree 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 lower-degree (two) basis of the top element as shown in the center 8.3. p- and hp-Re nement 15 Figure 8.2.15: Solution of Example 8.2.1 by uniform ( ) and adaptive ( ) h-re nement 33]. portion of the gure or by adding third- and fourth-order 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 lower-degree 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 lower-center 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 lower-order 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 piecewise-linear (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 \lower-order" piecewise 16 Adaptive Finite Element Techniques 10 1 1 0 1 0 1 0 1 20 1 0 1 10 1 0 1 0 1 0 1 40 1 0 1 0 1 0 1 10 2 1 0 1 0 1 0 1 02 1 0 1 40 1 0 1 40 1 0 1 0 1 0 1 40 1 01 1 0 1 0 1 0 1 02 1 0 1 01 1 0 1 0 1 0 1 04 1 0 1 0 1 0 1 01 10 1 2 1 0 1 0 1 0 1 0 1 0 2 1 0 1 0 1 1 0 1 0 1 02 1 0 1 0 1 0 1 0 1 0 1 40 1 0 1 0 1 0 1 40 1 02 1 0 1 20 1 0 1 40 1 0 1 0 1 0 1 40 1 1 0 1 01 1 0 1 0 1 04 1 0 1 0 1 0 1 01 10 1 1 0 1 0 1 0 1 20 1 0 1 0 1 10 1 0 1 0 1 40 1 0 1 0 1 0 1 10 2 1 0 1 0 1 0 1 04 1 0 1 40 1 0 1 40 1 0 1 0 1 0 1 40 1 01 1 0 1 0 1 0 1 02 1 0 1 0 1 01 1 0 1 0 1 04 1 0 1 0 1 0 1 01 Figure 8.3.1: Second- and fourth-degree hierarchical shape functions on two square elements are incompatible across the common...
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