# Spectral pollution and second-order relative spectra for self-adjoint operators

@article{Levitin2002SpectralPA, title={Spectral pollution and second-order relative spectra for self-adjoint operators}, author={Michael Levitin and Eugene Shargorodsky}, journal={Ima Journal of Numerical Analysis}, year={2002}, volume={24}, pages={393-416} }

We consider the phenomenon of spectral pollution arising in calculation of spectra of self-adjoint operators by projection methods. We suggest a strategy of dealing with spectral pollution by using the so-called second-order relative spectra. The effectiveness of the method is illustrated by a detailed analysis of two model examples.

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#### References

SHOWING 1-10 OF 16 REFERENCES

Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

- Mathematics
- 1998

This paper considers a number of related problems concerning the computation of eigenvalues and complex resonances of a general self-adjoint operator H . The feature which ties the different sections… Expand

A remark on spurious eigenvalues in a square

- Mathematics
- 1999

Abstract We study the finite element approximation of the eigensolutions of a second-order problem in a square arising in fluid-structure interaction. We analyze the schemes described in [1] and show… Expand

Spectral Pollution

- Mathematics
- 2002

M. PLUM Mathematisches Institut I, Universität Karlsruhe, Kaiserstrasse 12, D-76128 Karlsruhe, Germany [Received on 10 December 2002; revised on 12 January 2004] It is well known that routine methods… Expand

C*-Algebras and Numerical Linear Algebra

- Mathematics
- 1992

Given a self-adjoint operator A on a Hilbert space, suppose that one wishes to compute the spectrum of A numerically. In practice, these problems often arise in such a way that the matrix of A… Expand

On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form

- Computer Science, Mathematics
- Math. Comput.
- 2000

This paper presents examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues, and proves that bad behavior is proved analytically and demonstrated in numerical experiments. Expand

Analysis of Toeplitz Operators

- Mathematics
- 1991

A revised introduction to the advanced analysis of block Toeplitz operators including recent research. This book builds on the success of the first edition which has been used as a standard reference… Expand

On spectral pollution in the finite element approximation of thin elastic “membrane” shells

- Mathematics
- 1997

Summary.The bending terms in a shell are small with respect to membrane ones as the thickness tends to zero. Consequently, the membrane approximation gives a good description of vibration properties… Expand

THE SPECTRA OF TOEPLITZ'S MATRICES.*

- Mathematics
- 1950

(2) t~~~~~~~ (0) Y , fn e*. n=-co Let f (f-, f+), where f+ and f- are the respective sequences f+ = (fo, f,** f- = * , f-2., f4)., and let the corresponding functions f+ ((p) i f- (p) of class L2 be… Expand

Geometry of higher order relative spectra and projection methods

- Mathematics
- 2000

Let $H$ be a densely defined linear operator acting on a Hilbert space $\cH$, let $P$ be the orthogonal projection onto a closed linear subspace $\cL$ and let $n \in \bn$. The $n$-th order spectrum… Expand