211a since the h 1 norm dominates the l2 norm 7211a

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Unformatted text preview: kp+1: This estimate does not have an optimal rate since the interpolation error (7.2.10a) is converging as O(hp+1). Getting the correct rate for an L2 error estimate is more complicated than it is in H 1. The proof is divided into two parts. 7.2. Convergence and Optimality 9 Lemma 7.2.1. (Aubin-Nitsche) Under the assumptions of Theorem 7.2.3, let (x y) 2 H01 be the solution of the \dual problem" A(v ) = (v e) where 1 8v 2 H0 u;U e = ku ; U k : 0 (7.2.12a) (7.2.12b) N Let ; 2 S0 be an interpolant of , then ku ; U k0 ku ; U k1 k ; ;k1 : (7.2.12c) Proof. Set V = ; in (7.2.6) to obtain A(; u ; U ) = 0: (7.2.13) Take the L2 inner product of (7.2.12b) with u ; U to obtain ku ; U k0 = (e u ; U ): Setting v = u ; U in (7.2.12a) and using the above relation yields ku ; U k0 = A(u ; U ): Using (7.2.13) ku ; U k0 = A(u ; U ; ;): Now use the continuity of A(v u) in H 1 ((7.2.1) with s = 1) to obtain (7.2.12c). Since we have an estimate for ku ; U k1 , estimating ku ; U k0 by (7.2.12c) requires an estimate o...
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