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

D. Gallistl

Research output: Contribution to journalArticleAcademicpeer-review

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

Fingerprint

Rayleigh
Divergence
Polynomials
Approximation
Mesh Refinement
Piecewise Polynomials
Rayleigh-Ritz Method
Norm
Adaptive Mesh
Polytopes
Numerical Computation
Numerical Scheme
Eigenvalue Problem
Lipschitz
Convergence Rate
Error Estimates
Upper bound
Gradient
Eigenvalue
Computing

Keywords

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

Cite this

@article{3479a5ec234545f89384b8cea4e44557,
title = "Rayleigh-Ritz approximation of the inf-sup constant for the divergence",
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.",
keywords = "Cosserat spectrum, Inf-sup constant, LBB constant, Stokes system, noncompact eigen-value problem, upper bounds",
author = "D. Gallistl",
year = "2019",
month = "1",
doi = "10.1090/mcom/3327",
language = "English",
volume = "88",
pages = "73--89",
journal = "Mathematics of computation",
issn = "0025-5718",
publisher = "American Mathematical Society",
number = "315",

}

Rayleigh-Ritz approximation of the inf-sup constant for the divergence. / Gallistl, D.

In: Mathematics of computation, Vol. 88, No. 315, 01.2019, p. 73-89.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Rayleigh-Ritz approximation of the inf-sup constant for the divergence

AU - Gallistl, D.

PY - 2019/1

Y1 - 2019/1

N2 - 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.

AB - 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.

KW - Cosserat spectrum

KW - Inf-sup constant

KW - LBB constant

KW - Stokes system

KW - noncompact eigen-value problem

KW - upper bounds

UR - http://www.scopus.com/inward/record.url?scp=85053307053&partnerID=8YFLogxK

U2 - 10.1090/mcom/3327

DO - 10.1090/mcom/3327

M3 - Article

VL - 88

SP - 73

EP - 89

JO - Mathematics of computation

JF - Mathematics of computation

SN - 0025-5718

IS - 315

ER -