New expansions of numerical eigenvalues by finite elements

✍ Scribed by Hung-Tsai Huang; Zi-Cai Li; Qun Lin

• Book ID: 104005614
• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 239 KB
• Volume: 217
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2007.06.011

✦ Synopsis

The paper provides new expansions of leading eigenvalues foru = u in S with the Dirichlet boundary condition u = 0 on jS by finite elements, with the support of numerical experiments. The theoretical proof of new expansions of leading eigenvalues is given only for the bilinear element Q 1 . However, such a new proof technique can be applied to other elements, conforming and nonconforming. The new error expansions are reported for the Q 1 elements and other three nonconforming elements, the rotated bilinear element (denoted by Q rot 1 ), the extension of Q rot 1 (denoted by EQ rot 1 ) and Wilson's element. The expansions imply that Q 1 and Q rot 1 yield upper bounds of the eigenvalues, and that EQ rot 1 and Wilson's elements yield lower bounds of the eigenvalues. By the extrapolation, the O(h 4 ) convergence rate can be obtained, where h is the boundary length of uniform rectangles.