Well use examples to provide insight into the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online