Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian

An extra-stabilised Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplace Dirichlet eigenvalues. The smallness assumption $\min\{\lambda_h,\lambda\}h_{\max}^{4}$ $\le 184.9570$ in $2$D (resp. $\le 21.2912$ in $3$D) on the maximal mesh-size $h_{\max}$ makes the computed $k$-th discrete eigenvalue $\lambda_h\le \lambda$ a lower eigenvalue bound for the $k$-th Dirichlet eigenvalue $\lambda$. This holds for multiple and clusters of eigenvalues and serves for the localisation 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 $n\ge 2$, which are of independent interest. The convergence analysis in $3$D follows the Babu\v{s}ka-Osborn theory and relies on a companion operator for the Morley finite element method. This is based on the Worsey-Farin $3$D version of the Hsieh-Clough-Tocher macro element with a careful selection of center points in a further decomposition of each tetrahedron into $12$ sub-tetrahedra. Numerical experiments in $2$D support the optimal convergence rates of the extra-stabilised Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates.