Lower bounds of eigenvalues of the biharmonic operators by the rectangular Morley element methods
Provides rigorous lower bound guarantees for eigenvalue approximations in finite element methods, addressing a known gap for 3D biharmonic problems.
The paper proves that rectangular Morley element methods yield lower bounds for eigenvalues of biharmonic operators in 2D and 3D, resolving a difficulty in 3D via a novel decomposition. Numerical results confirm the theory.
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.