37 the rst term on the right is the standard

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Unformatted text preview: family of meshes h and u is smooth enough to be in H p+1. Brenner and Scott 12], Chapter 8, obtain similar results under similar conditions when interpolation is performed at the Lobatto points on the boundary of an element. The Lobatto polynomial of degree p is de ned on ;1 1] as p; 2 Lp( ) = dd p;2 (1 ; 2)p;1 2 ;1 1] p 2: These results are encouraging since the perturbation in the boundary data is of slightly higher order than the interpolation error. Unfortunately, if the domain is not smooth and, e.g., contains corners solutions will not be elements of H p+1. Less is known in these cases. 7.3.3 Perturbed Boundaries Suppose that the domain is replaced by a polygonal domain ~ as shown in Figure 7.3.1. Strang and Fix 18], analyze second-order problems with homogeneous Dirichlet data of the form: determine u 2 H01 satisfying A(v u) = (v f ) 1 8v 2 H0 (7.3.8a) 16 Analysis of the Finite Element Method where functions in H01 satisfy u(x y) = 0, (x y) 2 @ . The nite element solution ~N U 2 S0 satis es ~N A(V U ) = (V f ) 8V 2 S0 (7.3.8b) ~N ~N where functions in S0 vanish o...
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