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

In this article, we analyze the lower bound property of the discrete eigenvalues by the rectangular Morley elements of the biharmonic operators in both two dimension (2D) and three dimension (3D). 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 2D, and is negative and of second order for 3D, 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. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1623–1644, 2015