# Using 721 in conjunction with the rayleigh quotient

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Unformatted text preview: the Rayleigh quotient implies kur k2 s r 1: r kur k2 0 Combining the two results, kur k2 kur k2 s s r 1: r r k2 r k2 ku 0 ku 0 Thus, provides a lower bound for the minimum eigenvalue and provides a bound for the maximum growth rate of the eigenvalues in H s. Example 7.2.2. Solutions of the Dirichlet problem ;uxx ; uyy = f (x y ) (x y) 2 u=0 (x y) 2 @ satisfy the Galerkin problem (7.1.1) with A(v u) = ZZ rv rudxdy ru = ux uy ]T : An application of Cauchy's inequality reveals jA(v u)j = j ZZ rv rudxdy j krv k0 kruk0: 7.2. Convergence and Optimality where 5 kruk = 2 0 ZZ (u2 + u2 )dxdy: x y Since kruk0 kuk1, we have jA(v u)j kv k1 kuk1: Thus, (7.2.1) is satis ed with s = 1 and = 1, and the strain energy is continuous in H 1. Establishing that A(v u) is coercive in H 1 is typically done by using Friedrichs's rst inequality which states that there is a constant > 0 such that kruk2 0 kuk2 : 0 (7.2.3) Now, consider the identity A(u u) = kruk2 = (1=2)kruk2 + (1=2)kruk2 0 0 0 and use (7.2.3) to obtain A(u u) (1=2)kruk2 + (1=2) kuk2 0 0 kuk2...
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