Hence all terms in 733 or 735 have the same order of

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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 piecewise-linear (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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