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

Gallistl, Dietmar

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–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.

 Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2017 Sprache Englisch Identifikator ISSN: 2365-662X URN: urn:nbn:de:swb:90-711004 KITopen ID: 1000071100 Verlag KIT, Karlsruhe Umfang 17 S. Serie CRC 1173 ; 2017/15 Schlagworte inf-sup constant, LBB constant, Stokes system, non-compact eigenvalue problem, Cosserat spectrum, upper bounds
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