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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
Lp( ) = dd p;2 (1 ; 2)p;1
2 ;1 1]
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
U 2 S0 satis es
A(V U ) = (V f )
8V 2 S0
where functions in S0 vanish o...
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