In this talk we will present some recent progress in the analysis of the discrete eigenvalue produced by the nonconforming finite element methods for the symmetric elliptic eigenvalue problem. We obtained the principle of yielding lower bounds of eigenvalues for nonconforming finite element methods.This principle can be used as a guide to construct new nonconforming elements or modify the existing nonconforming elements to produce lower approximation. We prove the most used nonconforming element methods in the literature for various symmetric elliptic operators provide lower approximation. Some exceptions are also shown by the numerical experiments. Our analysis is valid for nonuniform grids,general elliptic operators,general boundary conditions and works for both smooth and non-smooth eigenfunctions.
This talk is based on the joint work with Hu Jun and Lin Qun.