[{"type":"report","title":"Rayleigh-Ritz approximation of the inf-sup constant for the divergence","issued":{"date-parts":[["2017"]]},"DOI":"10.5445\/IR\/1000071100","author":[{"family":"Gallistl","given":"Dietmar"}],"publisher":"KIT","publisher-place":"Karlsruhe","ISSN":"2365-662X","collection-title":"CRC 1173","abstract":"A numerical scheme for computing approximations to the inf-sup constant of the divergence operator in bounded Lipschitz polytopes in R$^{n}$ is proposed. The method is based on a conforming approximation of the pressure space based on piecewise polynomials of some fixed degree k $ \\geq \\ $0. The scheme can be viewed as a Rayleigh\u2013Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. The new approximation replaces the H\u207b\u00b9 norm of a gradient by a discrete H\u207b\u00b9 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 non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided.","keyword":"inf-sup constant, LBB constant, Stokes system, non-compact eigenvalue problem, Cosserat spectrum, upper bounds","number-of-pages":17,"kit-publication-id":"1000071100"}]