A convergent adaptive method for elliptic eigenvalue problems

Giani, S.; Graham, I.G.

doi:10.1137/070697264

A convergent adaptive method for elliptic eigenvalue problems

Giani, S.; Graham, I.G.

Authors

Dr Stefano Giani stefano.giani@durham.ac.uk
Associate Professor

I.G. Graham

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.

Citation

Giani, S., & Graham, I. (2009). A convergent adaptive method for elliptic eigenvalue problems. SIAM Journal on Numerical Analysis, 47(2), 1067-1091. https://doi.org/10.1137/070697264

Journal Article Type	Article
Publication Date	Jan 1, 2009
Deposit Date	Feb 11, 2013
Publicly Available Date	Feb 15, 2013
Journal	SIAM Journal on Numerical Analysis
Print ISSN	0036-1429
Electronic ISSN	1095-7170
Publisher	Society for Industrial and Applied Mathematics
Peer Reviewed	Peer Reviewed
Volume	47
Issue	2
Pages	1067-1091
DOI	https://doi.org/10.1137/070697264
Keywords	Second-order elliptic problems, Eigenvalues, Adaptive finite element methods, Convergence.