# Boundary layer tails in periodic homogenization

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 4, page 209-243
- ISSN: 1292-8119

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topAllaire, Grégoire, and Amar, Micol. "Boundary layer tails in periodic homogenization ." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 209-243. <http://eudml.org/doc/197276>.

@article{Allaire2010,

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.
},

author = {Allaire, Grégoire, Amar, Micol},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Boundary layers; periodic functions; asymptotic expansion;
homogenization.; rectangular domains having either fixed or oscillating boundary; homogenized equation; effective Fourier boundary conditions},

language = {eng},

month = {3},

pages = {209-243},

publisher = {EDP Sciences},

title = {Boundary layer tails in periodic homogenization },

url = {http://eudml.org/doc/197276},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Allaire, Grégoire

AU - Amar, Micol

TI - Boundary layer tails in periodic homogenization

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 209

EP - 243

AB -
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.

LA - eng

KW - Boundary layers; periodic functions; asymptotic expansion;
homogenization.; rectangular domains having either fixed or oscillating boundary; homogenized equation; effective Fourier boundary conditions

UR - http://eudml.org/doc/197276

ER -

## References

top- T. Abboud and H. Ammari, Diffraction at a curved grating, TM and TE cases, homogenization. J. Math. Anal. Appl.202 (1996) 995-1206. Zbl0865.35122
- Y. Achdou, 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. Zbl0800.78017
- Y. Achdou and O. Pironneau, Domain decomposition and wall laws. C.R. Acad. Sci. Paris Sér. I Math.320 (1995) 541-547. Zbl0834.76014
- 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. Zbl0931.35010
- G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures et Appl.77 (1998) 153-208. Zbl0901.35005
- G. Allaire and C. Conca, Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal.29 (1998) 343-379. Zbl0918.35018
- M. Avellaneda and F.-H. Lin, Homogenization of elliptic problems with Lp boundary data. Appl. Math. Optim.15 (1987) 93-107. Zbl0644.35034
- M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization. C.P.A.M., XL (1987) 803-847. Zbl0632.35018
- 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. Zbl0343.35022
- N. Bakhvalov and G. Panasenko, Homogenization, averaging processes in periodic media. Kluwer Academic Publishers, Dordrecht, Mathematics and its Applications36 (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). Zbl0404.35001
- A. Bensoussan, J.L. Lions and G. Papanicolaou, Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci.15 (1979) 53-157. Zbl0408.60100
- A. Bourgeat and E. Marusic-Paloka, Non-linear effects for flow in periodically constricted channel caused by high injection rate. Mathematical Models and Methods in Applied Sciences8 (1998) 379-405. Zbl0920.76082
- R. Brizzi and J.P. Chalot, Homogénéisation de frontière. PhD Thesis, Université de Nice (1978).
- G. Buttazzo and R.V. Kohn, Reinforcement by a thin layer with oscillating thickness. Appl. Math. Optim.16 (1987) 247-261.
- G. Chechkin, A. Friedman and A. Piatnitski, The boundary-value problem in domains with very rapidly oscillating boundary. INRIA Report 3062 (1996). Zbl0938.35049
- R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique, Tome 3, Masson, Paris (1984). Zbl0642.35001
- 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).
- A. Friedman, B. Hu and Y. Liu, A boundary value problem for the Poisson equation with multi-scale oscillating boundary. J. Diff. Eq.137 (1997) 54-93. Zbl0878.35014
- W. Jäger and A. Mikelic, 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. Zbl0878.76076
- E. Landis and G. Panasenko, 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. Zbl0378.35020
- J.L. Lions, Some methods in the mathematical analysis of systems and their controls. Science Press, Beijing, Gordon and Breach, New York (1981). Zbl0542.93034
- S. Moskow and M. Vogelius, First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof. Proc. Roy. Soc. Edinburg127 (1997) 1263-1295. Zbl0888.35011
- O. Oleinik, A. Shamaev and G. Yosifian, Mathematical problems in elasticity and homogenization. North Holland, Amsterdam (1992). Zbl0768.73003
- E. Sánchez-Palencia, Non homogeneous media and vibration theory. Springer Verlag, Lecture notes in physics 127 (1980). Zbl0432.70002
- F. Santosa and W. Symes, A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math.51 (1991) 984-1005. Zbl0741.73017
- F. Santosa and M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium. SIAM J. Appl. Math.53 (1993) 1636-1668. Zbl0808.35085

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