Direct Guaranteed Lower Eigenvalue Bounds with Optimal a Priori Convergence Rates for the Bi-Laplacian
Carsten Carstensen, Sophie Puttkammer
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 \(\min \{\lambda_h,\lambda \}h_{\max }^{4}\) \(\le 184.9570\) in 2D (resp., \(\le 21.2912\) in 3D) 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 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 \(n\ge 2\) , 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.Keywordsbiharmonic eigenvalue problemdirect guaranteed lower eigenvalue boundsMorley finite elementconforming companionnonconforming interpolationHsieh–Clough–TocherWorsey–Farina priori error estimatesadaptive mesh-refinementMSC codes65N2565N3065N15