### 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 |
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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 |

### Keywords

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

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## Cite this

*Mathematics of computation*,

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