A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids

Abstract

This paper is a generalization and improvement of some recent results concerned with the lower bound property of eigenvalues produced by the Morley element of the biharmonic operator. Such an extension is of two fold. First, we generalize those results for the biharmonic operator to general fourth order elliptic operators; second, we extend them from quasi-uniform grids to local quasi-uniform grids. In particular, for the general case, we prove the asymptotic lower bound property of the discrete eigenvalues under a very weak saturation condition in terms of local meshsizes which can in general not be improved; for the special case with B = 0, we show a similar result without any assumption except the fineness of the mesh. These results can be regarded as a substantial improvement of a related result on regular meshes, including adaptive local refined meshes, for general fourth order elliptic operators, due to Yang, Li and Bi [http://arxiv.org/abs/1509.00566], while the analysis herein is completely different and largely simpler than that in the above paper.