Unformatted text preview: ere N is the number of elements in the mesh and kE k2 is the restriction of the error
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
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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