Lower spectral bounds by Wilson's brick discretization (Q979219)

A nonconforming finite element, the Wilson element, is applied for solving the eigenvalue problem of the Laplace operator in polygonal domains in two and three space dimensions. A relationship between the interpolation weak estimate of the Wilson element and the n-linear element is established. The authors prove that the three-dimensional Wilson's brick eigenvalues approximate the exact eigenvalues from below, which is claimed to be a new progress for the finite element method. A numerical experiment for the eigenvalues of the Laplace operator in three dimensions, considering a unit cube, shows that the results given by the Wilson's brick yield indeed lower spectral bounds and thus confirm the theoretical results.