DIRECT GUARANTEED LOWER EIGENVALUE BOUNDS WITH OPTIMAL A PRIORI CONVERGENCE RATES FOR THE BI-LAPLACIAN

Abstract

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 min{λh, λ}h4max ≤ 184.9570 in 2D (resp., ≤ 21.2912 in 3D) on the maximal mesh-size hmax makes the computed kth discrete eigenvalue λh ≤ λ a lower eigenvalue bound for the kth Dirichlet eigenvalue λ. This holds for multiple and clusters of eigenvalues and serves for the localization of the bi-Laplacian Dirichlet eigenvalues, in particular for coarse meshes. The analysis requires interpolation error estimates for the Morley FEM with explicit constants in any space dimension n ≥ 2, which are of independent interest. The convergence analysis in 3D follows the Babuška-Osborn theory and relies on a companion operator for the Morley finite element method. This is based on the Worsey-Farin 3D version of the Hsieh-Clough-Tocher macro element with a careful selection of center points in a further decomposition of each tetrahedron into 12 subtetrahedra. Numerical experiments in 2D support the optimal convergence rates of the extra-stabilized Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates.