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December  2011, 6(4): 755-781. doi: 10.3934/nhm.2011.6.755

The needle problem approach to non-periodic homogenization

1. 

Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany

2. 

McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, H3A 2K6 Montreal QC, Canada

Received  January 2011 Revised  July 2011 Published  December 2011

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation $-\nabla\cdot (a^\epsilon\nabla u^\epsilon) = f$. On the coefficients $a^\epsilon$ we assume that solutions $u^\epsilon$ of homogeneous $\epsilon$-problems on simplices with average slope $\xi\in \mathbb{R}^n$ have the property that flux-averages $f a^\epsilon\nabla u^\epsilon\in \mathbb{R}^n$ converge, for $\epsilon\to 0$, to some limit $a^\star(\xi)$, independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an $\epsilon$-problem in each simplex and are affine on faces.
Citation: Ben Schweizer, Marco Veneroni. The needle problem approach to non-periodic homogenization. Networks & Heterogeneous Media, 2011, 6 (4) : 755-781. doi: 10.3934/nhm.2011.6.755
References:
[1]

G. Allaire, Homogenization of the Stokes flow in a connected porous medium,, Asymptotic Analysis, 2 (1989), 203. Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[3]

G. Allaire, Dispersive limits in the homogenization of the wave equation,, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 415. doi: 10.5802/afst.1055. Google Scholar

[4]

G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization,, Multiscale Model. Simul., 4 (2005), 790. doi: 10.1137/040611239. Google Scholar

[5]

I. Babuška, Homogenization and its application. Mathematical and computational problems,, in, (1976), 89. Google Scholar

[6]

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Homogenization in deterministic and stochastic problems,, in, (1977), 106. Google Scholar

[7]

G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry,, Multiscale Model. Simul., 8 (2010), 717. doi: 10.1137/09074557X. Google Scholar

[8]

A. Bourgeat, A. Mikelic and S. Wright, Stochastic two-scale convergence in the mean and applications,, J. Reine Angew. Math., 456 (1994), 19. Google Scholar

[9]

P. G. Ciarlet, "The Finite Element Method For Elliptic Problems,", Reprint of the 1978 original [North-Holland, 40 (1978). Google Scholar

[10]

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 99. Google Scholar

[11]

S. Conti and B. Schweizer, Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance,, Comm. Pure Appl. Math., 59 (2006), 830. doi: 10.1002/cpa.20115. Google Scholar

[12]

_____, A sharp-interface limit for a two-well problem in geometrically linear elasticity,, Arch. Ration. Mech. Anal., 179 (2006), 413. doi: 10.1007/s00205-005-0397-y. Google Scholar

[13]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization,, Ann. Mat. Pura Appl. (4), 144 (1986), 347. doi: 10.1007/BF01760826. Google Scholar

[14]

E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine,, Boll. Un. Mat. Ital. (4), 8 (1973), 391. Google Scholar

[15]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. Google Scholar

[16]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems,, J. Amer. Math. Soc., 18 (2005), 121. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[17]

T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media,, J. Comput. Phys., 134 (1997), 169. doi: 10.1006/jcph.1997.5682. Google Scholar

[18]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, "Homogenization Of Differential Operators And Integral Functionals,", Translated from the Russian by G. A. Yosifian, (1994). Google Scholar

[19]

S. M. Kozlov, The averaging of random operators,, Mat. Sb. (N.S.), 109(151) (1979), 188. Google Scholar

[20]

C. Melcher and B. Schweizer, Direct approach to $L^p$ estimates in homogenization theory,, Ann. Mat. Pura Appl. (4), 188 (2009), 399. doi: 10.1007/s10231-008-0078-1. Google Scholar

[21]

F. Murat and L. Tartar, $H$-convergence,, in, 31 (1997), 21. Google Scholar

[22]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043. Google Scholar

[23]

G. Nguetseng, Homogenization structures and applications. I,, Z. Anal. Anwendungen, 22 (2003), 73. doi: 10.4171/ZAA/1133. Google Scholar

[24]

H. Owhadi and L. Zhang, Metric-based upscaling,, Comm. Pure Appl. Math., 60 (2007), 675. doi: 10.1002/cpa.20163. Google Scholar

[25]

W. Rudin, "Real And Complex Analysis,", Third edition, (1987). Google Scholar

[26]

E. Sánchez-Palencia, "Nonhomogeneous Media And Vibration Theory,", Lecture Notes in Physics, 127 (1980). Google Scholar

[27]

B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping,, SIAM J. Math. Anal., 39 (2008), 1740. doi: 10.1137/060675472. Google Scholar

[28]

B. Schweizer and M. Veneroni, On non-periodic homogenization of time-dependent equations,, Submitted to Nonlinear Anal. B: Real World Appl., (). Google Scholar

[29]

_____, Periodic homogenization of the Prandtl-Reuss model with hardening,, J. Multiscale Modeling, 2 (2010), 69. doi: 10.1142/S1756973710000291. Google Scholar

[30]

L. Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles,, in, (1975), 420. Google Scholar

[31]

_____, "The General Theory Of Homogenization. A Personalized Introduction,", Lecture Notes of the Unione Matematica Italiana, 7 (2009). Google Scholar

[32]

M. Veneroni, Stochastic homogenization of subdifferential inclusions via scale integration,, Intl. J. of Struct. Changes in Solids, 3 (2011), 83. Google Scholar

[33]

S. Wright, On the steady-state flow of an incompressible fluid through a randomly perforated porous medium,, J. Differ. Equations, 146 (1998), 261. doi: 10.1006/jdeq.1998.3436. Google Scholar

[34]

V. V. Zhikov, Estimates for an averaged matrix and an averaged tensor,, Uspekhi Mat. Nauk, 46 (1991), 49. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization of the Stokes flow in a connected porous medium,, Asymptotic Analysis, 2 (1989), 203. Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[3]

G. Allaire, Dispersive limits in the homogenization of the wave equation,, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 415. doi: 10.5802/afst.1055. Google Scholar

[4]

G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization,, Multiscale Model. Simul., 4 (2005), 790. doi: 10.1137/040611239. Google Scholar

[5]

I. Babuška, Homogenization and its application. Mathematical and computational problems,, in, (1976), 89. Google Scholar

[6]

A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Homogenization in deterministic and stochastic problems,, in, (1977), 106. Google Scholar

[7]

G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry,, Multiscale Model. Simul., 8 (2010), 717. doi: 10.1137/09074557X. Google Scholar

[8]

A. Bourgeat, A. Mikelic and S. Wright, Stochastic two-scale convergence in the mean and applications,, J. Reine Angew. Math., 456 (1994), 19. Google Scholar

[9]

P. G. Ciarlet, "The Finite Element Method For Elliptic Problems,", Reprint of the 1978 original [North-Holland, 40 (1978). Google Scholar

[10]

D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 99. Google Scholar

[11]

S. Conti and B. Schweizer, Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance,, Comm. Pure Appl. Math., 59 (2006), 830. doi: 10.1002/cpa.20115. Google Scholar

[12]

_____, A sharp-interface limit for a two-well problem in geometrically linear elasticity,, Arch. Ration. Mech. Anal., 179 (2006), 413. doi: 10.1007/s00205-005-0397-y. Google Scholar

[13]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization,, Ann. Mat. Pura Appl. (4), 144 (1986), 347. doi: 10.1007/BF01760826. Google Scholar

[14]

E. De Giorgi and S. Spagnolo, Sulla convergenza degli integrali dell'energia per operatori ellittici del secondo ordine,, Boll. Un. Mat. Ital. (4), 8 (1973), 391. Google Scholar

[15]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. Google Scholar

[16]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems,, J. Amer. Math. Soc., 18 (2005), 121. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[17]

T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media,, J. Comput. Phys., 134 (1997), 169. doi: 10.1006/jcph.1997.5682. Google Scholar

[18]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, "Homogenization Of Differential Operators And Integral Functionals,", Translated from the Russian by G. A. Yosifian, (1994). Google Scholar

[19]

S. M. Kozlov, The averaging of random operators,, Mat. Sb. (N.S.), 109(151) (1979), 188. Google Scholar

[20]

C. Melcher and B. Schweizer, Direct approach to $L^p$ estimates in homogenization theory,, Ann. Mat. Pura Appl. (4), 188 (2009), 399. doi: 10.1007/s10231-008-0078-1. Google Scholar

[21]

F. Murat and L. Tartar, $H$-convergence,, in, 31 (1997), 21. Google Scholar

[22]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043. Google Scholar

[23]

G. Nguetseng, Homogenization structures and applications. I,, Z. Anal. Anwendungen, 22 (2003), 73. doi: 10.4171/ZAA/1133. Google Scholar

[24]

H. Owhadi and L. Zhang, Metric-based upscaling,, Comm. Pure Appl. Math., 60 (2007), 675. doi: 10.1002/cpa.20163. Google Scholar

[25]

W. Rudin, "Real And Complex Analysis,", Third edition, (1987). Google Scholar

[26]

E. Sánchez-Palencia, "Nonhomogeneous Media And Vibration Theory,", Lecture Notes in Physics, 127 (1980). Google Scholar

[27]

B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping,, SIAM J. Math. Anal., 39 (2008), 1740. doi: 10.1137/060675472. Google Scholar

[28]

B. Schweizer and M. Veneroni, On non-periodic homogenization of time-dependent equations,, Submitted to Nonlinear Anal. B: Real World Appl., (). Google Scholar

[29]

_____, Periodic homogenization of the Prandtl-Reuss model with hardening,, J. Multiscale Modeling, 2 (2010), 69. doi: 10.1142/S1756973710000291. Google Scholar

[30]

L. Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles,, in, (1975), 420. Google Scholar

[31]

_____, "The General Theory Of Homogenization. A Personalized Introduction,", Lecture Notes of the Unione Matematica Italiana, 7 (2009). Google Scholar

[32]

M. Veneroni, Stochastic homogenization of subdifferential inclusions via scale integration,, Intl. J. of Struct. Changes in Solids, 3 (2011), 83. Google Scholar

[33]

S. Wright, On the steady-state flow of an incompressible fluid through a randomly perforated porous medium,, J. Differ. Equations, 146 (1998), 261. doi: 10.1006/jdeq.1998.3436. Google Scholar

[34]

V. V. Zhikov, Estimates for an averaged matrix and an averaged tensor,, Uspekhi Mat. Nauk, 46 (1991), 49. Google Scholar

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