Lower bounds of eigenvalues of the biharmonic operators by the rectangular Morley element methods

TL;DR

This paper investigates the lower bounds of eigenvalues for biharmonic operators using rectangular Morley element methods, providing theoretical analysis and numerical validation in 2D and 3D.

Contribution

It introduces a new analysis of the lower bound property for discrete eigenvalues with rectangular Morley elements, including a novel decomposition technique and saturation condition.

Findings

Discrete eigenvalues are proven to be lower bounds of the exact eigenvalues.

The analysis reveals higher order error terms in 2D and negative second order terms in 3D.

Numerical results confirm the theoretical lower bound properties.

Abstract

In this paper, we analyze the lower bound property of the discrete eigenvalues by the rectangular Morley elements of the biharmonic operators in both two and three dimensions. The analysis relies on an identity for the errors of eigenvalues. We explore a refined property of the canonical interpolation operators and use it to analyze the key term in this identity. In particular, we show that such a term is of higher order for two dimensions, and is negative and of second order for three dimensions, which causes a main difficulty. To overcome it, we propose a novel decomposition of the first term in the aforementioned identity. Finally, we establish a saturation condition to show that the discrete eigenvalues are smaller than the exact ones. We present some numerical results to demonstrate the theoretical results.