Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



High order multi-scale wall-laws, Part I: The periodic case

Authors: Didier Bresch and Vuk Milisic
Journal: Quart. Appl. Math. 68 (2010), 229-253
MSC (2000): Primary 76D05, 35B27, 76Mxx, 65Mxx
DOI: https://doi.org/10.1090/S0033-569X-10-01135-0
Published electronically: March 10, 2010
MathSciNet review: 2663000
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Abstract: In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a Newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelić, W. Jäger, J. Diff. Eqs, 170, 96-122, (2001)] and [Y. Achdou et al, J. Comput. Phys., 147, 1, 187-218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to $ \epsilon$ (the roughness' thickness). We establish mathematically a poor convergence rate for averaged second order wall-laws as it was illustrated numerically for instance in [Y. Achdou, et al]. In comparison, we establish exponential error estimates in the case of an explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that their rate of convergence is at least of order $ \epsilon^{\frac{3}{2}}$. We provide a numerical assessment of the claims as well as a counterexample that makes evident the impossibility of an averaged second order wall-law. Our paper may be seen as the first one to derive efficient high order wall-laws boundary conditions.

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Additional Information

Didier Bresch
Affiliation: LAMA, UMR 5127 CNRS, Université de Savoie, 73217 Le Bourget du Lac cedex, France
Email: didier.bresch@univ-savoie.fr

Vuk Milisic
Affiliation: LJK-IMAG, UMR 5523 CNRS, 51 rue des Mathématiques, B.P.53, 38041 Grenoble cedex 9, France
Email: vuk.milisic@imag.fr

DOI: https://doi.org/10.1090/S0033-569X-10-01135-0
Keywords: Wall-laws, rough boundary, Laplace equation, multi-scale modelling, boundary layers, finite element methods, error estimates.
Received by editor(s): February 20, 2008
Published electronically: March 10, 2010
Additional Notes: The first author was partially supported by the project “Études mathématiques de paramétrisations en océanographie” that is part of the “ACI jeunes chercheurs 2004” framework of the French Research Ministry and by a Rhône Alpes project “Equations de type Saint-Venant avec viscosité pour des problèmes environnementaux”.
The second author was partially supported by a contract with Cardiatis$^{®}$, a company providing metallic multi-layer stents for cerebral and aortic aneurysms. This research has been partly funded by the Rhône-Alpes Institute of Complex Systems IXXI, http://www.ixxi.fr. The authors would like to thank E. Bonnetier for fruitful discussions and helpful proofreading.
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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