New expansions of numerical eigenvalues by finite elements

Abstract: The paper provides new expansions of leading eigenvalues for in S with the Dirichlet boundary condition on 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 . However, such a new proof technique can be applied to other elements, conforming and nonconforming. The new error expansions are reported for the elements and other three nonconforming elements, the rotated bilinear element (denoted by ), the extension of (denoted by ) and Wilson''s element. The expansions imply that and yield upper bounds of the eigenvalues, and that and Wilson''s elements yield lower bounds of the eigenvalues. By the extrapolation, the convergence rate can be obtained, where h is the boundary length of uniform rectangles. [Copyright &y& Elsevier]