11 this implies that 2 e ke k2 n where ke ke is the

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Unformatted text preview: f the enriched mesh. Using this notion, we divide the error estimate kE k2 by a factor n so that e kE k2 e n 2 ~ N : Thus, each element of the current mesh is divided into n segments such that n ~ N kE ke 2 : (8.2.6) In practice, n and N may be rounded up or increased slightly to provide a measure of assurance that the error criterion will be satis ed after the next adaptive solution. The mesh division process may be implemented by repeated applications of a meshre nement algorithm without solving the partial di erential equation in between. Thus, with bisection 34, 33], the elemental error estimate would be halved on each bisected element. Re nement would then be repeated until (8.2.6) is satis ed. The error estimation process (8.2.6) works with coarsening when n < 1 however, neighboring elements would have to suggest coarsening as well. Example 8.2.1 Rivara 33] solves Laplace's equation uxx + uyy = 0 (x y) 2 where is a regular hexagon inscribed in a unit circle. The hexagon is oriented with one vertex along the positive x-axis with a \crack" through this vertex for 0 x 1, y = 0. Boundary conditions are established to be homogeneous Neumann conditions on the x-axis below the crack and u(r ) = r1=4 sin 4 everywhere else. This function is also the exact solution of the problem expressed in a polar frame eminating from the center of the hexagon. The solution has a singularity at the origin due to the \re-entrant" angle of 2 at the crack tip and the change in 14 Adaptive Finite Element Techniques boundary conditions from Dirichlet to Neumann. The solution was computed with a piecewise-linear nite element basis using quasi-uniform and adaptive h-re nement. A residual error estimation procedure similar to those described in Section 7.4 was used to appraise solution accuracy 33]. Re nement followed (8.2.3). The results shown in Figure 8.2.15 indicate that the uniform mesh is converging as O(N ;1=8 ) where N is the number of degrees of freedom. We have seen (Section 7.2) that uniform h-re nement converges as kek1 C1hmin(p q) = C2 N ; min(p q)=2 (8.2.7) where q > 0 depends on the solution smoothness and, in two dimensions, N / h2 . For linear elliptic problems with geometric singularities, q = =! where ! is the maximum interior angle on @ . For the hexagon with a crack, the interior angles would be =3, 2 =3, and 2 . The latter is the largest angle hence, q = 1=2. Thus, with p = 1, convergence should occur at an O(N ;1=4) rate however, the actual rate is lower (Figure 8.2.15). The adaptive procedure has restored the O(N ;1=2 ) convergence rate that one would expect of a problem without singularities. In general, optimal adaptive h-re nement will converge as 6, 43] kek1 C1 hp = C2N ;p=2 : (8.2.8) 8.3 p- and hp-Re nement With p-re nement, the mesh is not changed but the order of the nite element basis is varied locally over the domain. As with h-re nement, we must ensure that the basis remains continuous at element boundaries. A situation where second- and fourth-degree hierarchical bases intersect along an edge between tw...
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