Rayleigh-Ritz approximation of the inf-sup constant for the divergence

D. Gallistl

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

Abstract

A numerical scheme for computing approximations to the inf-sup constant of the divergence operator in bounded Lipschitz polytopes in Rn is proposed. The method is based on a conforming approximation of the pressure space based on piecewise polynomials of some fixed degree k ≥ 0. The scheme can be viewed as a Rayleigh-Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. The new approximation replaces the H -1 norm of a gradient by a discrete H -1 norm which behaves monotonically under mesh refinement. By discretizing the pressure space with piecewise polynomials, upper bounds to the inf-sup constant are obtained. Error estimates are presented that prove convergence rates for the approximation of the inf-sup constant provided it is an isolated eigenvalue of the corresponding noncompact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided.

Original languageEnglish
Pages (from-to)73-89
Number of pages17
JournalMathematics of computation
Volume88
Issue number315
DOIs
Publication statusPublished - Jan 2019
Externally publishedYes

Keywords

  • Cosserat spectrum
  • Inf-sup constant
  • LBB constant
  • Stokes system
  • noncompact eigen-value problem
  • upper bounds

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