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

Gallistl, Dietmar

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–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 non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided.

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Volltext §
DOI: 10.5445/IR/1000071100
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2017
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000071100
Verlag KIT, Karlsruhe
Umfang 17 S.
Serie CRC 1173 ; 2017/15
Schlagwörter inf-sup constant, LBB constant, Stokes system, non-compact eigenvalue problem, Cosserat spectrum, upper bounds
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