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Subspace correction multi‐level methods for elliptic eigenvalue problems
Author(s) -
Chan Tony F.,
Sharapov Ilya
Publication year - 2001
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.238
Subject(s) - mathematics , rayleigh quotient , rayleigh quotient iteration , eigenvalues and eigenvectors , domain decomposition methods , schwarz alternating method , linear subspace , multiplicative function , additive schwarz method , subspace topology , convergence (economics) , computation , mathematical optimization , iterative method , algorithm , mathematical analysis , preconditioner , finite element method , pure mathematics , physics , quantum mechanics , economics , thermodynamics , economic growth
In this work, we apply the ideas of domain decomposition and multi‐grid methods to PDE‐based eigenvalue problems represented in two equivalent variational formulations. To find the lowest eigenpair, we use a “subspace correction” framework for deriving the multiplicative algorithm for minimizing the Rayleigh quotient of the current iteration. By considering an equivalent minimization formulation proposed by Mathew and Reddy, we can use the theory of multiplicative Schwarz algorithms for non‐linear optimization developed by Tai and Espedal to analyse the convergence properties of the proposed algorithm. We discuss the application of the multiplicative algorithm to the problem of simultaneous computation of several eigenfunctions also formulated in a variational form. Numerical results are presented. Copyright © 2001 John Wiley & Sons, Ltd.