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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