New expansions of numerical eigenvalues by Wilson's element

The paper explores new expansions of eigenvalues for −Δu=λρu in S with Dirichlet boundary conditions by Wilson's element. The expansions indicate that Wilson's element provides lower bounds of the eigenvalues. By the extrapolation or the splitting extrapolation, the O(h4) convergence rate can be obtained, where h is the maximal boundary length of uniform rectangles. Numerical experiments are carried to verify the theoretical analysis made. It is worth pointing out that these results are new, compared with the recent book, Lin and Lin [Q. Lin, J. Lin, Finite Element Methods; Accuracy and Improvement, Science Press, Beijing, 2006].