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 higherror 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, pre nement would be the preferred strategy. 18 Adaptive Finite Element Techniques
If e 1 the recommended strategy would be hre nement. Ai a 2] selects pre nement if e
and hre 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 hpre nement
for a variety of spatial error tolerances. The initial mesh for hre nement contained
32 triangular elements and used piecewisequadratic (p = 2) shape functions. For pre nement, the mesh contained 64 triangles with p varying from 1 to 5. The solution
with adaptive hpre 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 hre 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 hpre 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 hpRe 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 hpre nement ( , , and ., respectively). 20 Adaptive Finite Element Techniques...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Finite Element Method, coarsening, Adaptive Finite Element, Finite Element Techniques, J.E. Flaherty

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