2021/05/04 by Carsten Carstensen, Carstensen, Carsten, Sophie Puttkammer +1 · 1 citation
Computer Science · Engineering · Mathematics · #65N25 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2105.01505
openalex publication_date 2021/05/04 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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\λh,λ\hmax4 ≤ 184.9570 in 2D (resp. ≤ 21.2912 in 3D) on the maximal mesh-size hmax makes the computed k-th discrete eigenvalue λh≤ λ a lower eigenvalue bound for the k-th Dirichlet eigenvalue λ. 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≥ 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 sub-tetrahedra. Numerical experiments in 2D support the optimal convergence rates of the extra-stabilised Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates.