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Unformatted text preview: the Rayleigh quotient implies
Combining the two results,
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 =
0 ZZ (u2 + u2 )dxdy:
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
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
0 kuk2 :
0 (7.2.3) Now, consider the identity A(u u) = kruk2 = (1=2)kruk2 + (1=2)kruk2
and use (7.2.3) to obtain A(u u) (1=2)kruk2 + (1=2) kuk2
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14