×

Expository paper: A primer on homogenization of elliptic PDEs with stationary and ergodic random coefficient functions. (English) Zbl 1354.37085

Summary: We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.

MSC:

37N15 Dynamical systems in solid mechanics
37A05 Dynamical aspects of measure-preserving transformations
37A25 Ergodicity, mixing, rates of mixing
78A48 Composite media; random media in optics and electromagnetic theory
78M40 Homogenization in optics and electromagnetic theory
74Q99 Homogenization, determination of effective properties in solid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] A. Alexanderian, Random composite media: Homogenization, modeling, simulation, and material symmetry , Ph.D. thesis, University of Maryland, Baltimore County, 2010. · Zbl 1354.37085
[2] A. Alexanderian, M. Rathinam and R. Rostamian, Homogenization, symmetry, and periodization in diffusive random media , Acta Math. Sci. 32 (2012), 129-154. · Zbl 1265.78017 · doi:10.1016/S0252-9602(12)60008-3
[3] S.N. Armstrong, P. Cardaliaguet and P.E. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations , J. Amer. Math. Soc. 27 (2014), 479-540. · Zbl 1286.35023 · doi:10.1090/S0894-0347-2014-00783-9
[4] S.N. Armstrong and C.K. Smart, Stochastic homogenization of fully nonlinear uniformly elliptic equations revisited , Calc. Var. Part. Diff. Equat. (2012), 1-14.
[5] —-, Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity , Ann. Prob. 42 (2014), 2558-2594. · Zbl 1315.35019 · doi:10.1214/13-AOP833
[6] S.N. Armstrong and P.E. Souganidis, Stochastic homogenization of level-set convex Hamilton-Jacobi equations , Int. Math. Res. Not. IMRN 15 (2013), 3420-3449. · Zbl 1319.35003 · doi:10.1093/imrn/rns155
[7] V.I. Arnold and A. Avez, Ergodic problems of classical mechanics , W.A. Benjamin, Inc., New York, 1968. · Zbl 0715.70004
[8] A. Bensoussan, J.-L. Lions and G.C. Papanicolaou, Asymptotic analysis for periodic structures , Stud. Math. Appl. 5 , North-Holland Publishing Co., Amsterdam, 1978. · Zbl 0404.35001
[9] X. Blanc, R. Costaouec, C. Le Bris and F. Legoll, Variance reduction in stochastic homogenization using antithetic variables , Markov Proc. Rel. Fields 18 (2012), 31-66. · Zbl 1260.35260
[10] X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices , J. Math. Pure Appl. 88 (2007), 34-63. · Zbl 1129.60055 · doi:10.1016/j.matpur.2007.04.006
[11] A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization , Ann. Inst. H. Poin. Prob. Stat. 40 (2004), 153-165. · Zbl 1058.35023 · doi:10.1016/j.anihpb.2003.07.003
[12] M. Brin and G. Stuck, Introduction to dynamical systems , Cambridge University Press, Cambridge, 2002. · Zbl 1314.37002 · doi:10.1017/CBO9780511755316
[13] L.A. Caffarelli and P.E. Souganidis, Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media , Invent. Math. 180 (2010), 301-360. · Zbl 1192.35048 · doi:10.1007/s00222-009-0230-6
[14] L.A. Caffarelli, P.E. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media , Comm. Pure Appl. Math. 58 (2005), 319-361. · Zbl 1063.35025 · doi:10.1002/cpa.20069
[15] G.A. Chechkin, A.L. Piatnitski and A.S. Shamaev, Homogenization : Methods and applications , Trans. Math. Mono. 2345 , American Mathematical Society, Providence, RI, 2007. · Zbl 1128.35002
[16] G.H. Choe, Computational ergodic theory , Alg. Comp. Math. 13 , Springer-Verlag, Berlin, 2005. · Zbl 1064.37004 · doi:10.1007/b138894
[17] D. Cioranescu and P. Donato, An introduction to homogenization , Oxford Lect. Ser. Math. Appl. 17 , The Clarendon Press Oxford University Press, New York, 1999. · Zbl 0939.35001
[18] J. Conlon and T. Spencer, Strong convergence to the homogenized limit of elliptic equations with random coefficients , Trans. Amer. Math. Soc. 366 (2014), 1257-1288. · Zbl 1283.81102 · doi:10.1090/S0002-9947-2013-05762-5
[19] I.P. Cornfeld, S.V. Fomin and Y.G. Sinaĭ, Ergodic theory , Grundl. Math. Wissen. 245 , Springer-Verlag, New York, 1982.
[20] B. Dacorogna, Direct methods in the calculus of variations , Appl. Math. Sci. 78 , Springer, New York, 1989. · Zbl 0703.49001
[21] G. Dal Maso and L. Modica, Nonlinear stochastic homogenization , Ann. Mat. Pura Appl. 144 (1986), 347-389. · Zbl 0607.49010 · doi:10.1007/BF01760826
[22] —-, Nonlinear stochastic homogenization and ergodic theory , J. reine angew. Math. 368 (1986), 28-42. · Zbl 0582.60034
[23] Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations , SIAM J. Appl. Math. 65 (2004), 43-68. · Zbl 1088.65098 · doi:10.1137/S0036139903424886
[24] —-, Numerical homogenization of nonlinear random parabolic operators , Multiscale Model. Sim. 2 (2004), 237-268. · Zbl 1181.76113 · doi:10.1137/030600266
[25] A. Gloria, Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations , ESAIM: Math. Model. Numer. Anal. 46 (2012), 1-38. · Zbl 1282.35038 · doi:10.1051/m2an/2011018
[26] —-, Numerical homogenization: survey, new results, and perspectives , in ESAIM: Proceedings 37 (2012), 50-116. · Zbl 1329.65300 · doi:10.1051/proc/201237002
[27] A. Gloria, S. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization : Optimal bounds via spectral gap on glauber dynamics , Invent. Math. (2013), 1-61. · Zbl 1314.39020 · doi:10.1007/s00222-014-0518-z
[28] —-, A quantitative two-scale expansion in stochastic homogenization of discrete linear elliptic equations , Model. Math. Anal. Numer., 2013.
[29] A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations , Ann. Prob. 39 (2011), 779-856. · Zbl 1215.35025 · doi:10.1214/10-AOP571
[30] U. Hornung, Homogenization and porous media , Volume 6, Springer, 1997. · Zbl 0872.35002
[31] V.V. Jikov, S.M. Kozlov and O.A. Oleĭnik, Homogenization of differential operators and integral functionals , Springer-Verlag, Berlin, 1994 (in English).
[32] S.M. Kozlov, The averaging of random operators , Mat. Sb. (N.S.), 109 (1979), 188-202, 327.
[33] François Murat, Compacité par compensation , Ann. Scuol. Norm. Sup. Pisa 5 (1978), 489-507.
[34] O.A. Oleĭnik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization , Stud. Math. Appl. 26 , North-Holland Publishing Co., Amsterdam, 1992. · Zbl 0768.73003
[35] H. Owhadi, Approximation of the effective conductivity of ergodic media by periodization , Prob. Theor. Rel. Fields 125 (2003), 225-258. · Zbl 1040.60025 · doi:10.1007/s00440-002-0240-4
[36] A. Pankov, \(G\)-convergence and homogenization of nonlinear partial differential operators , Math. Appl. 422 , Kluwer Academic Publishers, Dordrecht, 1997. · Zbl 0883.35001
[37] G.C. Papanicolaou, Diffusion in random media , in Surveys Appl. Math. 1 , (1995), 205-253. · Zbl 0846.60081 · doi:10.1007/978-1-4899-0436-2_3
[38] G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients , in Random fields , Volumes I, II, Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 1981. · Zbl 0499.60059
[39] K. Sab, On the homogenization and the simulation of random materials , Europ. J. Mech. Solids 11 (1992), 585-607. · Zbl 0766.73008
[40] E. Sánchez-Palencia, Nonhomogeneous media and vibration theory , Lect. Notes Phys. 127 , Springer-Verlag, Berlin, 1980.
[41] L. Tartar, Compensated compactness and applications to partial differential equations , in Nonlinear analysis and mechanics 4 (1979), 136-211. · Zbl 0437.35004
[42] —-, The general theory of homogenization. A personalized introduction , Springer, Berlin, 2009. · Zbl 1188.35004 · doi:10.1007/978-3-642-05195-1
[43] S. Torquato, Random heterogeneous materials , Interdiscipl. Appl. Math. 16 , Springer-Verlag, New York, 2002. · Zbl 0988.74001
[44] P. Walters, An introduction to ergodic theory , Grad. Texts Math. 79 , Springer-Verlag, New York, 1982.
[45] V. Yurinskii, Averaging of symmetric diffusion in random medium , Siber. Math. J. 27 (1986), 603-613. · Zbl 0614.60051 · doi:10.1007/BF00969174
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.