Unformatted text preview: problem are displayed in Figure 1.3.4 for
a uniform mesh with eight elements. It appears that the pointwise discretization errors
are much smaller at nodes than they are globally. We'll see that this phenomena, called
superconvergence, applies more generally than this single example would imply.
Since nite element solutions are de ned as continuous functions (of x), we can also
appraise their behavior in some global norms in addition to the discrete error norms used
in Table 1.3.1. Many norms could provide useful information. One that we will use quite 18 Introduction
0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1.3.4: Exact and piecewise-linear nite element solutions of Example 1.3.1 on an
often is the square root of the strain energy of the error thus, using (1.3.2c) Z1
kekA := A(e e) =
p(e )2 + qe2 ]dx
0 0 1=2 : (1.3.23a) This expression may easily be evaluated as a summation over the elements in the spirit
of (1.3.9a). With p = q = 1 for this exampl...
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