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A posteriori error estimates which use the computed

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Unformatted text preview: estimates, which use the computed solution, provide more practical accuracy appraisals. 7.2 Convergence and Optimality While keeping the model problem (7.1.1) in mind, we will proceed in a slightly more general manner by considering a Galerkin problem of the form (7.1.1a) with a strain energy A(v u) that is a symmetric bilinear form (cf. De nitions 3.2.2, 3) and is also continuous and coercive. De nition 7.2.1. A bilinear form A(v u) is continuous in H s if there exists a constant > 0 such that jA(v u)j kukskv ks 8u v 2 H s : (7.2.1) De nition 7.2.2. A bilinear form A(u v) is coercive (H s ; elliptic or positive de nite) in H s if there exists a constant > 0 such that A(u u) kuk2 s 8u 2 H s : (7.2.2) Continuity and coercivity of A(v u) can be used to establish the existence and uniqueness of solutions to the Galerkin problem (7.1.1a). These results follow from the LaxMilgram Theorem. We'll subsequently prove a portion of this result, but more complete treatments appear elsewhere 6, 12, 13, 15]. We'll use examples to pro...
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