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21 with s 1 to obtain 7212c since we have an estimate

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Unformatted text preview: f k ; ;k1. This, of course, will be done by interpolation however, use of (7.2.10a) requires knowledge of the smoothness of . The following lemma provides the necessary a priori bound. Lemma 7.2.2. Let A(u v) be a symmetric, H 1-elliptic bilinear form and u be the solution of (7.2.8) on a smooth region . Then kuk2 C kf k0: (7.2.14) Remark 5. This result seems plausible since the underlying di erential equation is of second order, so the second derivatives should have the same smoothness as the righthand side f . The estimate might involve boundary data however, we have assumed trivial conditions. Let's further assume that @ E is not nil to avoid non-uniqueness issues. 10 Analysis of the Finite Element Method Proof. Strang and Fix 18], Chapter 1, establish (7.2.14) in one dimension. Johnson 14], Chapter 4, obtain a similar result. With preliminaries complete, here is the main result. Theorem 7.2.4. Given the assumptions of Theorem 7.2.3, then C hp+1kukp+1: ku ; U k0 (7.2.15) Proof. Applying (7.2.14) to the dual problem (7.2.1...
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