# A Two-Level Method for Nonsymmetric Eigenvalue Problems

A Two-Level Method for Nonsymmetric Eigenvalue Problems A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

# A Two-Level Method for Nonsymmetric Eigenvalue Problems

Acta Mathematicae Applicatae Sinica, Volume 21 (1) – Jan 1, 2005

## A Two-Level Method for Nonsymmetric Eigenvalue Problems

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 1–12 A Two-Level Method for Nonsymmetric Eigenvalue Problems Karel Kolman Mathematical Institute, Academy of Sciences of the Czech Republic, Zitn´ a 25, 115 67 Praha 1, Czech Republic (E-mail: kolman@math.cas.cz) Abstract A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin ﬁnite element discretization technique performed on a ﬁne grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the ﬁne grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived. Keywords eigenvalue problems; ﬁnite elements; postprocessing; two-level method; two-grid method; iterated Galerkin method 2000 MR Subject Classiﬁcation 65N15; 65N25; 65N30 1 Introduction [9] In 2001 Xu and Zhou described a two-grid method for computation of variationally formulated symmetric eigenvalue problems. The method solves an eigenvalue problem discretized on a coarse grid and obtains an improved solution by solving a linear problem on a ﬁne grid. Racheva [6] and Andreev altered the two-grid approach and used two-levels of discretization represented by ﬁnite element spaces of diﬀerent polynomial degrees. Both these papers deal mainly with the symmetric eigenvalue problem. Moreover, they do [3,8] not establish any links to the iterated Galerkin method described by Sloan .The relation between the two-level method and the iterated Galerkin method is in our opinion of vital importance. Therefore, this paper describes the two-level method in the context of the iterated Galerkin method and...

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Publisher
Springer Journals
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0209-z
Publisher site
See Article on Publisher Site

### Abstract

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

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