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

Authors:
Mats G. Larson
Affiliations:
-
Venue:
SIAM Journal on Numerical Analysis
Year:
2000

Citing 0
Cited 9

Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems

Journal of Computational and Applied Mathematics
A posteriori error estimates for the Steklov eigenvalue problem

Applied Numerical Mathematics
Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems

Journal of Computational and Applied Mathematics
$hp$-Finite Elements for Elliptic Eigenvalue Problems: Error Estimates Which Are Explicit with Respect to $\lambda$, $h$, and $p$

SIAM Journal on Numerical Analysis
Adaptive Discontinuous Galerkin Methods for Eigenvalue Problems Arising in Incompressible Fluid Flows

SIAM Journal on Scientific Computing
Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z2 Symmetry

Journal of Scientific Computing
Benchmark results for testing adaptive finite element eigenvalue procedures

Applied Numerical Mathematics
Robust A Priori and A Posteriori Error Analysis for the Approximation of Allen-Cahn and Ginzburg-Landau Equations Past Topological Changes

SIAM Journal on Numerical Analysis
A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement

Applied Numerical Mathematics

Quantified Score

Hi-index	0.01

Visualization

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.