Direct Guaranteed Lower Eigenvalue Bounds with Optimal a Priori Convergence Rates for the Bi-Laplacian

An extra-stabilized Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian Dirichlet eigenvalues. The smallness assumption in 2D (resp., 3D) on maximal mesh-size makes computed th discrete bound . This holds multiple and clusters of eigenvalues serves localization eigenvalues, particular coarse meshes. analysis requires interpolation error estimates FEM explicit constants any space dimension , which are independent interest. 3D follows Babuška–Osborn theory relies companion operator method. is based Worsey–Farin version Hsieh–Clough–Tocher macro careful selection center points further decomposition each tetrahedron into 12 subtetrahedra. Numerical experiments support suggest an adaptive algorithm empirical rates.