New expansions of numerical eigenvalues by finite elements

The paper provides new expansions of leading eigenvalues for -Δu=λρu 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 Q1. However, such a new proof technique can be applied to other elements, conforming and nonconforming. The new error expansions are reported for the Q1 elements and other three nonconforming elements, the rotated bilinear element (denoted by Q1rot), the extension of Q1rot (denoted by EQ1rot) and Wilson's element. The expansions imply that Q1 and Q1rot yield upper bounds of the eigenvalues, and that EQ1rot and Wilson's elements yield lower bounds of the eigenvalues. By the extrapolation, the O(h4) convergence rate can be obtained, where h is the boundary length of uniform rectangles.