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For the quadrature errors 736 to have the same ohp

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Unformatted text preview: )-plane should be = r1 + r = 2(p ; 1) + (p ; 1) = 3(p ; 1) 7.3. Perturbations 15 for (7.3.6a) and = r0 + r ; 1 = (3p ; 2) + (p ; 1) ; 1 = 4(p ; 1) for (7.3.6b). These results are to be compared with the order of 2(p ; 1) that was needed with the piecewise polynomials of degree p and linear transformations considered in Example 7.3.1. For quadratic transformations and approximations (p = 2), we need third- and fourth-order quadrature rules for O(h2) accuracy. 7.3.2 Interpolated Boundary Conditions Assume that integration is exact and the boundary @ is modeled exactly, but Dirichlet boundary data is approximated by a piecewise polynomial in S N , i.e., by a polynomial having the same degree p as the trial and test functions. Under these conditions, Wait and Mitchell 21], Chapter 6, show that the error in the solution U of a Galerkin problem with interpolated boundary conditions satis es ku ; U k1 C fhpkukp+1 + hp+1=2kukp+1g: (7.3.7) The rst term on the right is the standard interpolation error estimate. The second term corresponds to the perturbation due to approximating the boundary condition. As usual, computation is done on a uniform...
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