A Posteriori and a Priori Error Analysis for Finite Element Approximations of Self-Adjoint Elliptic Eigenvalue Problems

Abstract

We present a new error analysis for finite element approximations of self-adjoint elliptic eigenvalue problems. The analysis consists of three steps. First we prove a posteriori estimates for the error in the approximate eigenvectors and eigenvalues. The error in the eigenvectors is measured both in the L2 and energy norms. In these estimates the error is bounded in terms of the mesh size, a stability factor, and the residual, obtained by inserting the approximate eigenpair into the differential equation. The stability factor describes relevant stability properties of the continuous problem and we give a precise estimate of its size in terms of the spectrum of the continuous problem, the mesh size, and the choice of norm. Next we prove an a priori estimate of the residual in terms of derivatives of the exact eigenvectors and the mesh size. Finally we obtain precise a priori error estimates by combination of the a posteriori error estimates with the a priori residual estimate. The analysis shows that the a posteriori estimates are optimal and may be used for quantitative error estimation and design of adaptive algorithms.