In this form the constant c is independent of h and p

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Unformatted text preview: r to increase the polynomial degree when the solution u is smooth (q is large) and to reduce h near singularities. Thus, a possible strategy would be to increase p in smooth high-error regions and re ne the mesh near singularities. We, therefore, need a method of estimating solution smoothness and Ai a 2] does this by computing the ratio if e (p ; 1) 6= 0 e (p)= e (p ; 1) (8.3.5) e= 0 otherwise where p is the polynomial degree on Element e. An argument has been added to the error indicator on Element e to emphasize its dependence on the local polynomial degree. As described in Section 8.2, (p ; 1) can be estimated from the part of U involving the hierarchal corrections of degree p. Now If e < 1, the error estimate is decreasing with increasing polynomial degree. If enrichment were indicated on Element e, p-re nement would be the preferred strategy. 18 Adaptive Finite Element Techniques If e 1 the recommended strategy would be h-re nement. Ai a 2] selects p-re nement if e and h-re nement if e > , with 0:6. Adjustments have to made when p = 1 2]. Coarsening is done by vertex collapsing when all elements surrounding a vertex have low errors 2]. Example 8.3.1 Ai a 2] solves the nonlinear parabolic partial di erential equation ut ; u2(1 ; u) = uxx + uyy (x y) 2 t>0 2 with the initial and Dirichlet boundary data de ned so that the exact solution on the square = f(x y)j0 < x y < 2g is 1p p u(x y t) = =2(x+y;t =2) 1+e Although this problem is parabolic, Ai a 2] kept the temporal error small so that spatial errors dominate. Ai a 2] solved this problem with = 500 by adaptive h-, p-, and hp-re nement for a variety of spatial error tolerances. The initial mesh for h-re nement contained 32 triangular elements and used piecewise-quadratic (p = 2) shape functions. For pre nement, the mesh contained 64 triangles with p varying from 1 to 5. The solution with adaptive hp-re nement was initiated with 32 elements and p = 1, The convergence history of the three adaptive strategies is reported in Figure 8.3.2. The solution with h-re nement appears to be converging at an algebraic rate of approximately N ;0:95 , which is close to the theoretical rate (cf. (8.2.7)). There are no singularities in this problem and the adaptive p- and hp-re nement methods appear to be converging at exponential rates. This example and the material in this chapter give an introduction to the essential ideas of adaptivity and adaptive nite element analysis. At this time, adaptive software is emerging. Robust and reliable error estimation procedures are only known for model problems. Optimal enrichment strategies are just being discovered for realistic problems. 8.3. p- and hp-Re nement 19 0 10 Relative Error In H1 Norm −1 10 −2 10 −3 10 1 10 2 3 10 10 Number Of Degrees Of Freedom 4 10 Figure 8.3.2: Errors vs. the number of degrees of freedom N for Example 8.3.1 at t = 0:05 using adaptive h-, p- and hp-re nement ( , , and ., respectively). 20 Adaptive Finite Element Techniques...
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