### 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 language | English |
---|---|

Pages (from-to) | 73-89 |

Number of pages | 17 |

Journal | Mathematics of computation |

Volume | 88 |

Issue number | 315 |

DOIs | |

Publication status | Published - Jan 2019 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### Cite this

*Mathematics of computation*,

*88*(315), 73-89. https://doi.org/10.1090/mcom/3327

}

*Mathematics of computation*, vol. 88, no. 315, pp. 73-89. https://doi.org/10.1090/mcom/3327

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

Research output: Contribution to journal › Article › Academic › peer-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 -