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On the parameter choice in grad-div stabilization for incompressible flow problems

Please always quote using this URN:urn:nbn:de:0296-matheon-11917
  • Grad-div stabilization has been proved to be a very useful tool in discretizations of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. Depending on the situation, the optimal stabilization parameter might range from being very small to very large. The analytic results are supported by numerical examples.

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Metadaten
Author:Eleanor Jenkins, Volker John, Alexander Linke, Leo G. Rebholz
URN:urn:nbn:de:0296-matheon-11917
Referee:Alexander Mielke
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2013/01/08
Release Date:2013/01/08
Tag:error estimates; grad-div stabilization; incompressible Navier-Stokes equations; mixed finite elements; parameter choice
Institute:Research Center Matheon
Freie Universität Berlin
Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
MSC-Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Qxx Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05] / 35Q30 Navier-Stokes equations [See also 76D05, 76D07, 76N10]
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
76-XX FLUID MECHANICS (For general continuum mechanics, see 74Axx, or other parts of 74-XX) / 76Mxx Basic methods in fluid mechanics [See also 65-XX] / 76M10 Finite element methods
Preprint Number:995
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