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Ginzburg-Landau model with small pinning domains
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 and 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), 203-222. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [3] G. Allaire, Dispersive limits in the homogenization of the wave equation, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 415-431. doi: 10.5802/afst.1055. [4] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul., 4 (2005), 790-812 (electronic). doi: 10.1137/040611239. [5] I. Babuška, Homogenization and its application. Mathematical and computational problems, in "Numerical Solution of Partial Differential Equations, III" (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), Academic Press, New York, (1976), 89-116. [6] A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Homogenization in deterministic and stochastic problems, in "Stochastic Problems In Dynamics" (Sympos., Univ. Southampton, Southampton, 1976), Pitman, London, (1977), 106-115. [7] G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750. doi: 10.1137/09074557X. [8] A. Bourgeat, A. Mikelic and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. [9] P. G. Ciarlet, "The Finite Element Method For Elliptic Problems," Reprint of the 1978 original [North-Holland, Amsterdam], Classics in Applied Mathematics, 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. [10] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104. [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), 830-868. doi: 10.1002/cpa.20115. [12] _____, A sharp-interface limit for a two-well problem in geometrically linear elasticity, Arch. Ration. Mech. Anal., 179 (2006), 413-452. doi: 10.1007/s00205-005-0397-y. [13] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), 144 (1986), 347-389. doi: 10.1007/BF01760826. [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), 391-411. [15] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), 87-132. [16] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156 (electronic). doi: 10.1090/S0894-0347-04-00469-2. [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), 169-189. doi: 10.1006/jcph.1997.5682. [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, Springer-Verlag, Berlin, 1994. [19] S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109(151) (1979), 188-202, 327. [20] C. Melcher and B. Schweizer, Direct approach to $L^p$ estimates in homogenization theory, Ann. Mat. Pura Appl. (4), 188 (2009), 399-416. doi: 10.1007/s10231-008-0078-1. [21] F. Murat and L. Tartar, $H$-convergence, in "Topics in the Mathematical Modelling of Composite Materials," Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 21-43. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [23] G. Nguetseng, Homogenization structures and applications. I, Z. Anal. Anwendungen, 22 (2003), 73-107. doi: 10.4171/ZAA/1133. [24] H. Owhadi and L. Zhang, Metric-based upscaling, Comm. Pure Appl. Math., 60 (2007), 675-723. doi: 10.1002/cpa.20163. [25] W. Rudin, "Real And Complex Analysis," Third edition, McGraw-Hill Book Co., New York, 1987. [26] E. Sánchez-Palencia, "Nonhomogeneous Media And Vibration Theory," Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980. [27] B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping, SIAM J. Math. Anal., 39 (2008), 1740-1763. doi: 10.1137/060675472. [28] B. Schweizer and M. Veneroni, On non-periodic homogenization of time-dependent equations, Submitted to Nonlinear Anal. B: Real World Appl. [29] _____, Periodic homogenization of the Prandtl-Reuss model with hardening, J. Multiscale Modeling, 2 (2010), 69-106. doi: 10.1142/S1756973710000291. [30] L. Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles, in "Control Theory, Numerical Methods and Computer Systems Modelling" (Internat. Sympos., IRIA LABORIA, Rocquencourt, 1974), Lecture Notes in Econom. and Math. Systems, Vol. 107, Springer, Berlin, (1975), 420-426. [31] _____, "The General Theory Of Homogenization. A Personalized Introduction," Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, UMI, Bologna, 2009. [32] M. Veneroni, Stochastic homogenization of subdifferential inclusions via scale integration, Intl. J. of Struct. Changes in Solids, 3 (2011), 83-98. [33] S. Wright, On the steady-state flow of an incompressible fluid through a randomly perforated porous medium, J. Differ. Equations, 146 (1998), 261-286. doi: 10.1006/jdeq.1998.3436. [34] V. V. Zhikov, Estimates for an averaged matrix and an averaged tensor, Uspekhi Mat. Nauk, 46 (1991), 49-109, 239.

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##### References:
 [1] G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymptotic Analysis, 2 (1989), 203-222. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [3] G. Allaire, Dispersive limits in the homogenization of the wave equation, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 415-431. doi: 10.5802/afst.1055. [4] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul., 4 (2005), 790-812 (electronic). doi: 10.1137/040611239. [5] I. Babuška, Homogenization and its application. Mathematical and computational problems, in "Numerical Solution of Partial Differential Equations, III" (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), Academic Press, New York, (1976), 89-116. [6] A. Bensoussan, J.-L. Lions, and G. C. Papanicolaou, Homogenization in deterministic and stochastic problems, in "Stochastic Problems In Dynamics" (Sympos., Univ. Southampton, Southampton, 1976), Pitman, London, (1977), 106-115. [7] G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750. doi: 10.1137/09074557X. [8] A. Bourgeat, A. Mikelic and S. Wright, Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51. [9] P. G. Ciarlet, "The Finite Element Method For Elliptic Problems," Reprint of the 1978 original [North-Holland, Amsterdam], Classics in Applied Mathematics, 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. [10] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104. [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), 830-868. doi: 10.1002/cpa.20115. [12] _____, A sharp-interface limit for a two-well problem in geometrically linear elasticity, Arch. Ration. Mech. Anal., 179 (2006), 413-452. doi: 10.1007/s00205-005-0397-y. [13] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization, Ann. Mat. Pura Appl. (4), 144 (1986), 347-389. doi: 10.1007/BF01760826. [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), 391-411. [15] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), 87-132. [16] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156 (electronic). doi: 10.1090/S0894-0347-04-00469-2. [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), 169-189. doi: 10.1006/jcph.1997.5682. [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, Springer-Verlag, Berlin, 1994. [19] S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109(151) (1979), 188-202, 327. [20] C. Melcher and B. Schweizer, Direct approach to $L^p$ estimates in homogenization theory, Ann. Mat. Pura Appl. (4), 188 (2009), 399-416. doi: 10.1007/s10231-008-0078-1. [21] F. Murat and L. Tartar, $H$-convergence, in "Topics in the Mathematical Modelling of Composite Materials," Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, (1997), 21-43. [22] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [23] G. Nguetseng, Homogenization structures and applications. I, Z. Anal. Anwendungen, 22 (2003), 73-107. doi: 10.4171/ZAA/1133. [24] H. Owhadi and L. Zhang, Metric-based upscaling, Comm. Pure Appl. Math., 60 (2007), 675-723. doi: 10.1002/cpa.20163. [25] W. Rudin, "Real And Complex Analysis," Third edition, McGraw-Hill Book Co., New York, 1987. [26] E. Sánchez-Palencia, "Nonhomogeneous Media And Vibration Theory," Lecture Notes in Physics, 127, Springer-Verlag, Berlin-New York, 1980. [27] B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping, SIAM J. Math. Anal., 39 (2008), 1740-1763. doi: 10.1137/060675472. [28] B. Schweizer and M. Veneroni, On non-periodic homogenization of time-dependent equations, Submitted to Nonlinear Anal. B: Real World Appl. [29] _____, Periodic homogenization of the Prandtl-Reuss model with hardening, J. Multiscale Modeling, 2 (2010), 69-106. doi: 10.1142/S1756973710000291. [30] L. Tartar, Problèmes de contrôle des coefficients dans des équations aux dérivées partielles, in "Control Theory, Numerical Methods and Computer Systems Modelling" (Internat. Sympos., IRIA LABORIA, Rocquencourt, 1974), Lecture Notes in Econom. and Math. Systems, Vol. 107, Springer, Berlin, (1975), 420-426. [31] _____, "The General Theory Of Homogenization. A Personalized Introduction," Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag, Berlin, UMI, Bologna, 2009. [32] M. Veneroni, Stochastic homogenization of subdifferential inclusions via scale integration, Intl. J. of Struct. Changes in Solids, 3 (2011), 83-98. [33] S. Wright, On the steady-state flow of an incompressible fluid through a randomly perforated porous medium, J. Differ. Equations, 146 (1998), 261-286. doi: 10.1006/jdeq.1998.3436. [34] V. V. Zhikov, Estimates for an averaged matrix and an averaged tensor, Uspekhi Mat. Nauk, 46 (1991), 49-109, 239.
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