Theorem 724 given the assumptions of theorem 723 then

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Unformatted text preview: 2a) yields k k2 C kek0 = C since kek0 = 1 according to (7.2.12b). With 2 H2, we may use (7.2.10c) with q = s = 1 to obtain k ; ;k1 C hk k2 = Ch: Combining this estimate with (7.2.11a) and (7.2.12c) yields (7.2.15). Problems 1. Show that the function u that minimizes = w 2H A(w w) kwk0 6=0 (w w ) min 1 0 is u1, the eigenfunction corresponding to the minimum eigenvalue (v u). 1 of A(v u) = 2. Assume that A(v u) is a symmetric, continuous, and H 1-elliptic bilinear form and, for simplicity, that u v 2 H01. 2.1. Show that the strain energy and H 1 norms are equivalent in the sense that kuk2 1 A(u u) kuk2 1 1 8u 2 H0 : where and satisfy (7.2.1) and (7.2.2). 2.2. Prove Theorem 7.2.1. 3. Prove Corollary 7.2.1 to Theorem 7.2.2. 7.3 Perturbations In this section, we examine the e ects of perturbations due to numerical integration, interpolated boundary conditions, and curved boundaries. 7.3. Perturbations 7.3.1 11 Quadrature Perturbations With numerical integration, we determine U as the solution of N 8V 2 S0 A (V U ) = (V f ) (7.3.1a) instead of d...
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