Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-24T23:01:49.256Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundary layer tails in periodic homogenization

Published online by Cambridge University Press:  15 August 2002

Grégoire Allaire  and
Micol Amar
Grégoire Allaire
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Paris VI, Tour 55–65, 4 place Jussieu, 70005 Paris, France; allaire@ann.jussieu.fr.
Micol Amar
Affiliation:
Dipartimento Me.Mo.Mat, Università “La Sapienza" - Roma, Via A. Scarpa 16, 00161 Roma, Italy; amar@dmmm.uniroma1.it.
Get access

Abstract

This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly oscillating near the boundary and decay exponentially fast in the interior to a non-zero limit that we call boundary layer tail. The influence of these boundary layer tails on interior error estimates is emphasized. They mainly have two effects (at first order with respect to the period ε): first, they add a dispersive term to the homogenized equation, and second, they yield an effective Fourier boundary condition.

Keywords

Boundary layersperiodic functionsasymptotic expansionhomogenization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 209 - 243
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abboud, T. and Ammari, H., Diffraction at a curved grating, TM and TE cases, homogenization. J. Math. Anal. Appl. 202 (1996) 995-1206. CrossRef
Achdou, Y., Effect d'un revêtement métallisé mince sur la réflexion d'une onde électromagnétique. C.R. Acad. Sci. Paris Sér. I Math. 314 (1992) 217-222.
Achdou, Y. and Pironneau, O., Domain decomposition and wall laws. C.R. Acad. Sci. Paris Sér. I Math. 320 (1995) 541-547.
Y. Achdou and O. Pironneau, A 2nd order condition for flow over rough walls, in Proc. Int. Conf. on Nonlinear Diff. Eqs. and Appl., Bangalore, Shrikant Ed. (1996).
G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport. M2AN to appear.
Allaire, G. and Conca, C., Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures et Appl. 77 (1998) 153-208. CrossRef
Allaire, G. and Conca, C., Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal. 29 (1998) 343-379. CrossRef
Avellaneda, M. and Lin, F.-H., Homogenization of elliptic problems with Lp boundary data. Appl. Math. Optim. 15 (1987) 93-107. CrossRef
M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization. C.P.A.M., XL (1987) 803-847.
I. Babuška, Solution of interface problems by homogenization I, II, III. SIAM J. Math. Anal. 7 (1976) 603-634 and 635-645; 8 (1977) 923-937. CrossRef
N. Bakhvalov and G. Panasenko, Homogenization, averaging processes in periodic media. Kluwer Academic Publishers, Dordrecht, Mathematics and its Applications 36 (1990).
G. Bal, First-order corrector for the homogenization of the criticality eigenvalue problem in the even parity formulation of the neutron transport, to appear.
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North Holland, Amsterdam (1978).
Bensoussan, A., Lions, J.L. and Papanicolaou, G., Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci. 15 (1979) 53-157. CrossRef
Bourgeat, A. and Marusic-Paloka, E., Non-linear effects for flow in periodically constricted channel caused by high injection rate. Mathematical Models and Methods in Applied Sciences 8 (1998) 379-405. CrossRef
R. Brizzi and J.P. Chalot, Homogénéisation de frontière. PhD Thesis, Université de Nice (1978).
Buttazzo, G. and Kohn, R.V., Reinforcement by a thin layer with oscillating thickness. Appl. Math. Optim. 16 (1987) 247-261. CrossRef
G. Chechkin, A. Friedman and A. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary. INRIA Report 3062 (1996).
R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique, Tome 3, Masson, Paris (1984).
B. Engquist and J.C. Nédélec, Effective boundary conditions for accoustic and electro-magnetic scaterring in thin layers. Internal report 278, CMAP École Polytechnique (1993).
Friedman, A., Hu, B. and Liu, Y., A boundary value problem for the Poisson equation with multi-scale oscillating boundary. J. Diff. Eq. 137 (1997) 54-93. CrossRef
Jäger, W. and Mikelic, A., On the boundary conditions at the contact interface between a porous medium and a free fluid. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 23 (1996) 403-465.
Landis, E. and Panasenko, G., A theorem on the asymptotics of solutions of elliptic equations with coefficients periodic in all variables except one. Soviet Math. Dokl. 18 (1977) 1140-1143.
J.L. Lions, Some methods in the mathematical analysis of systems and their controls. Science Press, Beijing, Gordon and Breach, New York (1981).
Moskow, S. and Vogelius, M., First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof. Proc. Roy. Soc. Edinburg 127 (1997) 1263-1295. CrossRef
O. Oleinik, A. Shamaev and G. Yosifian, Mathematical problems in elasticity and homogenization. North Holland, Amsterdam (1992).
E. Sánchez-Palencia, Non homogeneous media and vibration theory. Springer Verlag, Lecture notes in physics 127 (1980).
Santosa, F. and Symes, W., A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51 (1991) 984-1005. CrossRef
Santosa, F. and Vogelius, M., First-order corrections to the homogenized eigenvalues of a periodic composite medium. SIAM J. Appl. Math. 53 (1993) 1636-1668. CrossRef