New expansions of numerical eigenvalues by finite elements

The paper provides new expansions of leading eigenvalues for - Δ u = λ ρ u in S with the Dirichlet boundary condition u = 0 on ∂ S 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 1 rot ), the extension of Q 1 rot (denoted by EQ 1 rot ) and Wilsonʹs element. The expansions imply that Q 1 and Q 1 rot yield upper bounds of the eigenvalues, and that EQ 1 rot and Wilsonʹs 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.