Lower Bounds of Eigenvalues of the Stokes Operator by Nonconforming Finite Elements on Local Quasi-Uniform Grids

Abstract

This paper is a generalization of some recent results concerned with the lower bound property of eigenvalues produced by both the enriched rotated $Q_1$ and Crouzeix-Raviart elements of the Stokes eigenvalue problem. The main ingredient is a novel and sharp $L^2$ error estimate of discrete eigenfunctions, and a new error analysis of nonconforming finite element methods.