A Convergent Adaptive Method for Elliptic Eigenvalue Problems

Authors:
S. Giani;I. G. Graham
Affiliations:
Stefano.Giani@nottingham.ac.uk;I.G.Graham@bath.ac.uk
Venue:
SIAM Journal on Numerical Analysis
Year:
2009

Citing 0
Cited 5

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
An iterative adaptive finite element method for elliptic eigenvalue problems

Journal of Computational and Applied Mathematics
A framework for robust eigenvalue and eigenvector error estimation and Ritz value convergence enhancement

Applied Numerical Mathematics
Efficient application of nonlinear stationary operators in adaptive wavelet methods--the isotropic case

Numerical Algorithms

Quantified Score

Hi-index	0.00

Visualization

Abstract

We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in $H^1$. Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.