Smoothed aggregation for Helmholtz problems

Luke N. Olson, Jacob B. Schroder

Research output: Contribution to journalArticle

Abstract

We outline a smoothed aggregation algebraic multigrid method for 1D and 2D scalar Helmholtz problems with exterior radiation boundary conditions. We consider standard 1D finite difference discretizations and 2D discontinuous Galerkin discretizations. The scalar Helmholtz problem is particularly difficult for algebraic multigrid solvers. Not only can the discrete operator be complex-valued, indefinite, and non-self-adjoint, but it also allows for oscillatory error components that yield relatively small residuals. These oscillatory error components are not effectively handled by either standard relaxation or standard coarsening procedures. We address these difficulties through modifications of SA and by providing the SA setup phase with appropriate wave-like near null-space candidates. Much is known a priori about the character of the near null-space, and our method uses this knowledge in an adaptive fashion to find appropriate candidate vectors. Our results for GMRES preconditioned with the proposed SA method exhibit consistent performance for fixed points-per-wavelength and decreasing mesh size.

Original languageEnglish (US)
Pages (from-to)361-386
Number of pages26
JournalNumerical Linear Algebra with Applications
Volume17
Issue number2-3
DOIs
StatePublished - Apr 1 2010

Keywords

  • Algebraic multigrid (AMG)
  • Discontinuous galerkin
  • Helmholtz
  • Indefinite
  • Non-symmetric
  • Smoothed aggregation (SA)

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics

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