Numerical methods for spectral problems: theory and applications

P000008

Filtered subspace iteration for selfadjoint operator eigenvalue problems

*Jeffrey Ovall (Portland State University)
Jay Gopalakrishnan (Portland State University)
Grubisic Luka (University of Zagreb)
Parker Benjamin (Portland State University)

We consider filtered subspace iteration for approximating a cluster of
eigenvalues (and its associated eigenspace) of a (possibly
unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated
by a quadrature approximation of an operator-valued contour integral
of the resolvent. Resolvents on infinite dimensional spaces are
discretized in computable finite-dimensional spaces before the
algorithm is applied. This study focuses on how such
discretizations result in errors in the eigenspace approximations
computed by the algorithm. The computed eigenspace is then used to
obtain approximations of the eigenvalue cluster. Bounds for the
Hausdorff distance between the computed and exact eigenvalue
clusters are obtained in terms of the discretization parameters
within an abstract framework. Realizations of the proposed
approach for a model second-order elliptic operator using both a standard
finite element discretization and a discontinuous Petrov-Galerkin discretization
of the resolvent are considered both theoretically and empirically.

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Keywords

Eigenvalue problems, subspace iteration, finite element methods