American Institute of Mathematical Sciences

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October  2016, 9(5): 1575-1590. doi: 10.3934/dcdss.2016064

Homogenization: In mathematics or physics?

Shixin Xu 1, , Xingye Yue 1, and  Changrong Zhang 2,

1. 

Department of Mathematics, Soochow University, Suzhou 215006, China
2. 

High Speed Aerodynamics Institute, China Aerodynamisc Development and Research Center, Mianyang 622661, China

Received  November 2014 Revised  September 2015 Published  October 2016

In mathematics, homogenization theory considers the limitations of the sequences of the problems and their solutions when a parameter tends to zero. This parameter is regarded as the ratio of the characteristic size between the micro scale and macro scale. So what is considered is a sequence of problems in a fixed domain while the characteristic size in micro scale tends to zero. But in the real physics or engineering situations, the micro scale of a medium is fixed and can not be changed. In the process of homogenization, it is the size in macro scale which becomes larger and larger and tends to infinity. We observe that the homogenization in physics is not equivalent to the homogenization in mathematics up to some simple rescaling. With some direct error estimates, we explain in what sense we can accept the homogenized problem as the limitation of the original real physical problems. As a byproduct, we present some results on the mathematical homogenization of some problems with source term being only weakly compacted in $H^{-1}$, while in standard homogenization theory, the source term is assumed to be at least compacted in $H^{-1}$. A real example is also given to show the validation of our observation and results.
Keywords: unit transformation., Homogenization, re-scaling.
Mathematics Subject Classification: Primary: 35B27, 35B40; Secondary: 35M3.
Citation: Shixin Xu, Xingye Yue, Changrong Zhang. Homogenization: In mathematics or physics?. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1575-1590. doi: 10.3934/dcdss.2016064
References:
[1]

G. Allaire, Homogenization et convergence a deux echelles, application a un probleme de convection diffusion. C.R.Acad. Sci. Paris, 312 (1991), 581-586.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[3]

I. Babuška, Solution of problem with interfaces and singularities, in Mathematical aspects of finite elements in partial differential equations, C. de Boor ed., Academic Press, New York, (1974), 213-277. Google Scholar

[4]

I. Babuška, Homogenization approach in engineering, Lecture notes in economics and mathematical systems, M. Beckman and H. P. Kunzi(eds.), Springer-Verlag, 134 (1976), 137-153.  Google Scholar

[5]

I. Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III, Academic Press, (1976), 89-116.  Google Scholar

[6]

I. Babuška, The computational aspects of the homogenization problem, Computing methods in applied sciences and engineering, I, Lecture notes in mathematics, Springer-Verlag,Berlin Heidelberg New York, 704 (1976), 309-316.  Google Scholar

[7]

A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.  Google Scholar

[8]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford university press, 1999.  Google Scholar

[9]

S. M. Kozlov, The averaging of random operators, Mat.Sb.(N.S), 109 (1979), 188-202,327.  Google Scholar

[10]

K. Lichtenecker, Die dielektrizitätskonstante natürlicher und künstlicher mischkörper, Phys. Zeitschr., XXVII (1926), 115-158. Google Scholar

[11]

J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed. , Clarendon Press, Oxford, 1881. Google Scholar

[12]

F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, 31 (1997), 21-43. doi: 10.1007/978-1-4612-2032-9_3.  Google Scholar

[13]

G. Nguestseng, 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.  Google Scholar

[14]

O. A. Olenik and A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in mathematics and its applications, J.L. Lions, G.Papanicolaou, H. Fujita, H.B. Keller, 26, North-Holland, 1992.  Google Scholar

[15]

S. Poisson, Second mémoire sur la théorie du magnétisme, Mem. Acad. France 5, 1822. Google Scholar

[16]

S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all' equatione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699.  Google Scholar

[17]

T. A. Suslina, Homogenization of a stationary periodic maxwell system, St. Petersburg Math. J., 16 (2005), 863-922. doi: 10.1090/S1061-0022-05-00883-6.  Google Scholar

[18]

L. Tartar, Compensated compactness and partial differential equations, in Nolinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman, 39 (1979), 136-212.  Google Scholar

[19]

L. Tartar, H-measure, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), 193-230. doi: 10.1017/S0308210500020606.  Google Scholar

[20]

T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci., 9 (2011), 1163-1176. doi: 10.4310/CMS.2011.v9.n4.a12.  Google Scholar

[21]

V. V. Zhikov, Some estimates from homogenization theory, (Russian) Dokl. Akad. Nauk, 406 (2006), 597-601.  Google Scholar

[22]

V. V. Zhikov and O. A. Oleinik, Homogenization and G-convergence of differential operators, Russ. Math. Surv., 34 (1979), 65-147. Google Scholar

[23]

V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization et convergence a deux echelles, application a un probleme de convection diffusion. C.R.Acad. Sci. Paris, 312 (1991), 581-586.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[3]

I. Babuška, Solution of problem with interfaces and singularities, in Mathematical aspects of finite elements in partial differential equations, C. de Boor ed., Academic Press, New York, (1974), 213-277. Google Scholar

[4]

I. Babuška, Homogenization approach in engineering, Lecture notes in economics and mathematical systems, M. Beckman and H. P. Kunzi(eds.), Springer-Verlag, 134 (1976), 137-153.  Google Scholar

[5]

I. Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III, Academic Press, (1976), 89-116.  Google Scholar

[6]

I. Babuška, The computational aspects of the homogenization problem, Computing methods in applied sciences and engineering, I, Lecture notes in mathematics, Springer-Verlag,Berlin Heidelberg New York, 704 (1976), 309-316.  Google Scholar

[7]

A. Bensousan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.  Google Scholar

[8]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, 17, Oxford university press, 1999.  Google Scholar

[9]

S. M. Kozlov, The averaging of random operators, Mat.Sb.(N.S), 109 (1979), 188-202,327.  Google Scholar

[10]

K. Lichtenecker, Die dielektrizitätskonstante natürlicher und künstlicher mischkörper, Phys. Zeitschr., XXVII (1926), 115-158. Google Scholar

[11]

J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd Ed. , Clarendon Press, Oxford, 1881. Google Scholar

[12]

F. Murat and L. Tartar, H-convergence, Topics in the Mathematical Modelling of Composite Materials, 31 (1997), 21-43. doi: 10.1007/978-1-4612-2032-9_3.  Google Scholar

[13]

G. Nguestseng, 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.  Google Scholar

[14]

O. A. Olenik and A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in mathematics and its applications, J.L. Lions, G.Papanicolaou, H. Fujita, H.B. Keller, 26, North-Holland, 1992.  Google Scholar

[15]

S. Poisson, Second mémoire sur la théorie du magnétisme, Mem. Acad. France 5, 1822. Google Scholar

[16]

S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all' equatione del calore, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 657-699.  Google Scholar

[17]

T. A. Suslina, Homogenization of a stationary periodic maxwell system, St. Petersburg Math. J., 16 (2005), 863-922. doi: 10.1090/S1061-0022-05-00883-6.  Google Scholar

[18]

L. Tartar, Compensated compactness and partial differential equations, in Nolinear Analysis and Mechanics: Heriot-Watt Symposium, Pitman, 39 (1979), 136-212.  Google Scholar

[19]

L. Tartar, H-measure, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh, 115 (1990), 193-230. doi: 10.1017/S0308210500020606.  Google Scholar

[20]

T. Yu and X. Yue, Residual-free bubble methods for numerical homogenization of elliptic problems, Commun. Math. Sci., 9 (2011), 1163-1176. doi: 10.4310/CMS.2011.v9.n4.a12.  Google Scholar

[21]

V. V. Zhikov, Some estimates from homogenization theory, (Russian) Dokl. Akad. Nauk, 406 (2006), 597-601.  Google Scholar

[22]

V. V. Zhikov and O. A. Oleinik, Homogenization and G-convergence of differential operators, Russ. Math. Surv., 34 (1979), 65-147. Google Scholar

[23]

V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994. doi: 10.1007/978-3-642-84659-5.  Google Scholar

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