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Galerkin Eigenvector Approximations

Christopher Beattie

TL;DR

Both orthogonal-Galerkin and Petrov- Galerkin methods are considered here with a special emphasis on nonselfadjoint problems, thus extending earlier studies by Chatelin, Babuska and Osborn, and Knyazev.

Abstract

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace -- and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed and an application to Krylov subspace methods for large scale matrix eigenvalue problems is presented. New lower bounds to the $sep$ of a pair of operators are developed as well.

Galerkin Eigenvector Approximations

TL;DR

Both orthogonal-Galerkin and Petrov- Galerkin methods are considered here with a special emphasis on nonselfadjoint problems, thus extending earlier studies by Chatelin, Babuska and Osborn, and Knyazev.

Abstract

How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace -- and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed and an application to Krylov subspace methods for large scale matrix eigenvalue problems is presented. New lower bounds to the of a pair of operators are developed as well.

Paper Structure

This paper contains 19 sections, 19 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Setting of the Problem
  3. Operators defined via quadratic forms
  4. The gap
  5. The eigenvalue problem
  6. The Galerkin Method
  7. Discretization
  8. Projections
  9. Convergence
  10. Bounded $A$
  11. Basic Estimates
  12. Related Estimates and Interpretation
  13. Unbounded $A$ -- Estimates in ${\cal V}$
  14. Basic results
  15. Elliptic boundary value problems: Finite Elements
  16. ...and 4 more sections

Key Result

Lemma 3.1

If $Z$ is a bounded (nonorthogonal) projection on a Hilbert space ${\cal W}$, then $\|I-Z\|_{\cal W}=\|Z\|_{\cal W}$. Furthermore, if $\Pi$ denotes the ${\cal W}$-orthogonal projection onto $Ran(Z)$,

Theorems & Definitions (19)

  • Lemma 3.1
  • Theorem 3.2: Descloux, et al. Des1978Des1981
  • Theorem 3.3: Descloux, et al. Des1978Des1981
  • Theorem 3.4: Babuška and Osborn BabOsb
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 5.1
  • Lemma 5.2
  • ...and 9 more