Morley Finite Element Method for the Eigenvalues of the Biharmonic Operator

TL;DR

This paper introduces new error estimates and convergence results for the Morley finite element method applied to the eigenvalues of the biharmonic operator, improving understanding of its accuracy and efficiency.

Contribution

It develops a new $C^1$ conforming companion operator and provides eigenvalue error estimates that relate the eigenvalue errors to invariant subspace approximation errors.

Findings

Established $L^2$ error estimates without extra regularity assumptions.

Proved optimal convergence rates for adaptive Morley FEM for eigenvalue clusters.

Bound eigenvalue errors by invariant subspace approximation errors.

Abstract

This paper studies the nonconforming Morley finite element approximation of the eigenvalues of the biharmonic operator. A new $C^1$ conforming companion operator leads to an $L^2$ error estimate for the Morley finite element method which directly compares the $L^2$ error with the error in the energy norm and, hence, can dispense with any additional regularity assumptions. Furthermore, the paper presents new eigenvalue error estimates for nonconforming finite elements that bound the error of (possibly multiple or clustered) eigenvalues by the approximation error of the computed invariant subspace. An application is the proof of optimal convergence rates for the adaptive Morley finite element method for eigenvalue clusters.