Lower spectral bounds by Wilson's brick discretization

Abstract

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 Wilson's brick eigenvalues approximate the exact eigenvalues from below, and thereby make a new progress on such an open problem in the finite element method.