The lower bound property of the Morley element eigenvalues

TL;DR

This paper proves that Morley element eigenvalues provide lower bounds for the true eigenvalues in fourth-order elliptic problems, including adaptive meshes, and demonstrates their application in plate vibration analysis.

Contribution

It establishes the lower bound property of Morley element eigenvalues for general meshes and applies adaptive computation to practical vibration problems.

Findings

Morley eigenvalues approximate true eigenvalues from below on regular and adaptive meshes.

The method is effective for fourth-order elliptic eigenvalue problems with clamped boundary conditions.

Adaptive computation yields reliable lower bounds in vibration analysis.

Abstract

In this paper, we prove that the Morley element eigenvalues approximate the exact ones from below on regular meshes, including adaptive local refined meshes, for the fourth-order elliptic eigenvalue problems with the clamped boundary condition in any dimension. And we implement the adaptive computation to obtain lower bounds of the Morley element eigenvalues for the vibration problem of clamped plate under tension.