Well use examples to provide insight into the

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Unformatted text preview: vide insight into the meanings of continuity and coercivity. Example 7.2.1. Consider the variational eigenvalue problem: determine nontrivial u 2 H01 and 2 0 1) satisfying A(u v) = (u v) 1 8v 2 H0 : When A(v u) is the strain energy for the model problem (7.1.1), smooth solutions of this variational problem also satisfy the di erential eigenvalue problem ;(pux)x ; (puy )y + qu = u u=0 (x y ) 2 @ E un = 0 where n is the unit outward normal to @ . (x y ) 2 (x y) 2 @ N: 4 Analysis of the Finite Element Method Letting r and ur , r 1, be an eigenvalue-eigenfunction pair and using the variational statement with v = u = ur , we obtain the Rayleigh quotient rr = A(u u ) r 1: r (ur ur ) Since this result holds for all r, we have A(ur ur ) 1 = min r 1 (ur ur ) where 1 is the minimum eigenvalue. (As indicated in Problem 1, this result can be extended.) Using the Rayleigh quotient with (7.2.2), we have kur k2 s r 1: r kur k2 0 Since kur ks kur k0, we have r >0 r 1: Thus, r , r 1, and, in particular, 1. Using (7.2.1) in conjunction with...
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