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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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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14