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11 where n is the number of elements in the mesh and

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Unformatted text preview: ere N is the number of elements in the mesh and kE k2 is the restriction of the error e estimate kE k2 to Element e. The most popular method of determining where adaptivity is needed is to use kE ke as an enrichment indicator. Thus, we assume that large errors come from regions where the local error estimate kE ke is large and this is where we should re ne or concentrate the mesh and/or increase the method order. Correspondingly, the mesh would be coarsened or the polynomial degree of the basis lowered in regions where kE ke is small. This is the strategy that we'll follow (cf. Section 8.2) however, we reiterate that there is no proof of the optimality of enrichment in the vicinity of the largest local error estimate. Enrichment indicators other than local error estimates have been tried. The use of solution gradients is popular. This is particularly true of uid dynamics problems where error estimates are not readily available 14, 16, 17, 19]. In this chapter, we'll examine h-, p-, and hp-re nement. Strategies using r-re nement will be addressed in Chapter 9. 8.2 h-Re nement Mesh re nement strategies for elliptic (steady) problems need not consider coarsening. We can re ne an initially coarse mesh until the requested accuracy is obtained. This strategy might not be optimal and won't be, for example, if the coarse mesh is too ne in some regions. Nevertheless, we'll concentrate on re nement at the expense of 8.2. h-Re nement 3 coarsening. We'll also focus on two-dimensional problems to avoid the complexities of three-dimensional geometry. 8.2.1 Structured Meshes Let us rst consider adaptivity on structured meshes and then examine unstructuredmesh re nement. Re nement of an element of a structured quadrilateral-element mesh by bisection requires mesh lines running to the boundaries to retain the four-neighbor structure (cf. the left of Figure 8.2.1). This strategy is simple to implement and has been used with nite di erence computation 42] however, it clearly re nes many more elements than necessary. The customary way of avoiding the excess re nement is to introduce irregular nodes where the edges of a re ned element meet at the midsides of a coarser one (cf. the right of Figure 8.2.1). The mesh is no longer structured and our standard method of basis construction would create discontinuities at the irregular nodes. Figure 8.2.1: Bisection of an element of a structured rectangular-element mesh creating mesh lines running between the boundaries (left). The mesh lines are removed by creating irregular nodes (right). The usual strategy of handling continuity at irregular nodes is to constrain the basis. Let us illustrate the technique for a piecewise-bilinear basis. The procedure for higherorder piecewise polynomials is similar. Thus, consider an edge between Vertices 1 and 2 containing an irregular node 3 as shown in Figure 8.2.2. For simplicity, assume that the elements are h h squares and that those adjacent to Edge 1-2 are indexed 1, 2, and 3 as shown in the gure. For convenien...
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