Lower spectral bounds by Wilsonʹs brick discretization

This paper discusses the Wilson element approximation for the eigenvalue problem of Laplace operator on n-dimensional polygonal domain ( n = 2 , 3 ) , and the main results are as follows: (1) We establish the relationship between the interpolation weak estimate of the Wilson element and the interpolation weak estimate of n-linear element. (2) We prove that 3-dimensional Wilsonʹs brick eigenvalues approximate the exact eigenvalues from below, and thereby make a new progress on such an open problem in the finite element method.