Unformatted text preview: d, as is the case here, the Jacobian is constant, this result is equivalent to integrating the di erentiated terms in the strain energy exactly (cf., e.g.,
(7.1.1c)). 14 Analysis of the Finite Element Method
If > 2(p ; 1) so that r > p ; 1 then the error in integration is higher order than
the O(hp) interpolation error however, the interpolation error dominates and
ku ; U k1 = O(hp): The extra e ort in performing the numerical integration more accurately is not
justi ed.
If < 2(p ; 1) so that r < p ; 1 then the integration error dominates the interpolation error and determines the order of accuracy as
ku ; U k1 = O(h ;p+2): p ; 2.
Let us conclude this example by examining convergence rates for piecewiselinear (or
bilinear) approximations (p = 1). In this case, r1 = 0, r0 = 1, and r = . Interpolation
errors converge as O(h). The optimal order of accuracy of the quadrature rule is = 0,
i.e., only constant functions need be integrated exactly. Performing the integration more
accurately yie...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Finite Element Method, KU

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