Eigenvalue approximation from below using non-conforming finite elements

Abstract

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators, with emphasis on obtaining lower bounds. In addition, this article also contains some new materials for eigenvalue approximations of the Laplace operator, which include: 1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below; 2) the proof of the fact that the non-conforming EQ 1 rot element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements; 3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q 1 rot element approximate the true eigenvalues from below for the L-shaped domain. Finally, we list several unsolved problems.