Algebraic multigrid for higher-order finite elements

J. J. Heys, T. A. Manteuffel, Steve F. McCormick, L. N. Olson

Research output: Contribution to journalArticle

Abstract

Two related approaches for solving linear systems that arise from a higher-order finite element discretization of elliptic partial differential equations are described. The first approach explores direct application of an algebraic-based multigrid method (AMG) to iteratively solve the linear systems that result from higher-order discretizations. While the choice of basis used on the discretization has a significant impact on the performance of the solver, results indicate that AMG is capable of solving operators from both Poisson's equation and a first-order system least-squares (FOSLS) formulation of Stoke's equation in a scalable manner, nearly independent of basis order, p, for 3 < p ≤ 8. The second approach incorporates preconditioning based on a bilinear finite element mesh overlaying the entire set of degrees of freedom in the higher-order scheme. AMG is applied to the operator based on bilinear finite elements and is used as a preconditioner in a conjugate gradient (CG) iteration to solve the algebraic system derived from the high-order discretization. This approach is also nearly independent of p. Although the total iteration count is slightly higher than using AMG accelerated by CG directly on the high-order operator, the preconditioned approach has the advantage of a straightforward matrix-free implementation of the high-order operator, thereby avoiding typically large computational and storage costs.

Original languageEnglish (US)
Pages (from-to)520-532
Number of pages13
JournalJournal of Computational Physics
Volume204
Issue number2
DOIs
StatePublished - Apr 10 2005
Externally publishedYes

Keywords

  • Algebraic multigrid
  • Finite elements
  • Higher-order
  • Multigrid
  • Poisson
  • Stokes

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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