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

Author: Dietmar Gallistl
Journal: Math. Comp. 88 (2019), 73-89
MSC (2010): Primary 65N12, 65N15, 65N30, 76D07
Published electronically: March 29, 2018
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Abstract: A numerical scheme for computing approximations to the inf-sup constant of the divergence operator in bounded Lipschitz polytopes in $ \mathbb{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 noncompact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided.

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Additional Information

Dietmar Gallistl
Affiliation: Institut für Angewandte und Numerische Mathematik, Karlsruher Institut für Technologie, 76128 Karlsruhe, Germany
Address at time of publication: Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Keywords: Inf-sup constant, LBB constant, Stokes system, noncompact eigenvalue problem, Cosserat spectrum, upper bounds
Received by editor(s): June 23, 2017
Received by editor(s) in revised form: September 14, 2017
Published electronically: March 29, 2018
Additional Notes: This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
Article copyright: © Copyright 2018 American Mathematical Society