# 31 second and fourth degree hierarchical shape

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Unformatted text preview: edge between elements (left). This can be corrected by removing the third- and fourth-degree edge functions from the lower element (center) or by adding third- and fourth-degree edge functions to the upper element (right). The maximum degree of the shape function associated with a mesh entity is shown in each case. constant (p = 0) solution by using the value of the piecewise-linear solution at the center of Element e. The di erence between these two \solutions" furnishes an error estimate which, when used with the error estimate for the piecewise-linear solution, is linearly extrapolated to higher values of p. Having estimates of discretization errors as a function of p on each element, we can use a strategy similar to (8.2.6) to select a value of p to reduce the error on an element to its desired level. Often, however, a simpler strategy is used. As indicated earlier, the error estimate kE ke should be of size =N on each element of the mesh. When enrichment is indicated, e.g., when kE k > , we can increase the degree of the polynomial representation by one on any element e where e > R N: (8.3.1a) The parameter e is an enrichment indicator on Element e, which may be kE ke, and 1:1. If coarsening is done, the degree of the approximation on Element e can be R reduced by one when < C he N (8.3.1b) where he is the longest edge of Element e and C 0:1. The convergence rate of the p version of the nite element method is exponential when the solution has no singularities. For problems with singularities, p-re nement converges e 8.3. p- and hp-Re nement 17 as C N ;q kek (8.3.2) where q > 0 depends on the solution smoothness 22, 23, 24, 25, 26]. (The parameter q is intended to be generic and is not necessarily the same as the one appearing in (8.2.7)). With singularities, the performance of the p version of the nite element method depends on the mesh. Performance will be better when the mesh has been graded near the singularity. This suggests combining h- and p-re nement. Indeed, when proper mesh re nement is combined with an increase of the polynomial degree p, the convergence rate is exponential kek C e;q1N 2 q (8.3.3) where q1 and q2 are positive constants that depend on the smoothness of the exact solution and the nite element mesh. Generating the correct mesh is crucial and its construction is only known for model problems 22, 23, 24, 25, 26]. Oden et al. 30] developed a strategy for hp-re nement that involved taking three solution steps followed by an extrapolation. Some techniques do not attempt to adjust the mesh and the order at the same time, but, rather, adjust either the mesh or the order. We'll illustrate one of these, but rst cite the more explicit version of the error estimate (8.2.7) given by Babuska and Suri 7] kek1 C h pq min(p q) kukmin(p q)+1 : (8.3.4) The mesh must satisfy the uniformity condition, the polynomial-degree is uniform, and u 2 H q+1. In this form, the constant C is independent of h and p. This result and the previous estimates indicate that it is bette...
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