Skip to main content

Advertisement

SpringerLink
Log in

Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L ∞(Ω), \({\Omega \subset \mathbb R^d}\)) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general case of arbitrary bounded coefficients. For such problems, we introduce explicit and optimal finite dimensional approximations of solutions that can be viewed as a theoretical Galerkin method with controlled error estimates, analogous to classical homogenization approximations. In particular, this approach allows one to analyze a given medium directly without introducing the mathematical concept of an \({\epsilon}\) family of media as in classical homogenization. We define the flux norm as the L 2 norm of the potential part of the fluxes of solutions, which is equivalent to the usual H 1-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (for example, piecewise polynomial). We refer to this property as the transfer property. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities. These inequalities play the same role in our approach as the div-curl lemma in classical homogenization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alessandrini G., Nesi V.: Univalent σ-harmonic mappings: connections with quasiconformal mappings. J. Anal. Math. 90, 197–215 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Allaire G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Allaire G.: Shape optimization by the homogenization method. Applied Mathematical Sciences, Vol. 146. Springer, New York (2002)

    Google Scholar 

  4. Allaire G., Brizzi R.: A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4(3), 790–812 (2005) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ancona A.: Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. Nagoya Math. J. 165, 123–158 (2002)

    MATH  MathSciNet  Google Scholar 

  6. Arbogast T., Boyd K.J.: Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44(3), 1150–1171 (2006) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  7. Arbogast T., Huang C.-S., Yang S.-M.: Improved accuracy for alternating-direction methods for parabolic equations based on regular and mixed finite elements. Math. Models Methods Appl. Sci. 17(8), 1279–1305 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Babuška I., Caloz G., Osborn J.E.: Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31(4), 945–981 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Babuška I., Lipton R., Stuebner M.: The penetration function and its application to microscale problems. BIT 48(2), 167–187 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Babuška I., Osborn J.E.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20(3), 510–536 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  11. Babuška I., Osborn J.E.: Can a finite element method perform arbitrarily badly?. Math. Comp. 69(230), 443–462 (2000)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Bakhvalov N., Panasenko G.: Homogenization: averaging processes in periodic media. Mathematics and its Applications, Vol. 36. Kluwer, Dordrecht (1990)

    Google Scholar 

  13. Bensoussan A., Lions J.L., Papanicolaou G.: Asymptotic Analysis for Periodic Structure. North Holland, Amsterdam (1978)

    Google Scholar 

  14. Bernardi C., Verfürth R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Blanc X., Le Bris C., Lions P.-L.: Une variante de la théorie de l’homogénéisation stochastique des opérateurs elliptiques. C. R. Math. Acad. Sci. Paris 343(11–12), 717–724 (2006)

    MATH  MathSciNet  Google Scholar 

  16. Blanc X., Le Bris C., Lions P.-L.: Stochastic homogenization and random lattices. J. Math. Pures Appl. 9 88(1), 34–63 (2007)

    MATH  MathSciNet  Google Scholar 

  17. Braides A.: Γ-convergence for beginners. Lecture Series in Mathematics and its Applications, Vol. 22. Oxford University Press, Oxford (2002)

    Google Scholar 

  18. Branets L.V., Ghai S.S., L. L., Wu X.-H.: Challenges and technologies in reservoir modeling. Commun. Comput. Phys. 6(1), 1–23 (2009)

    Article  MathSciNet  Google Scholar 

  19. Brenner S.C., Scott L.R.: The mathematical theory of finite elements methods. Texts in Applied Mathematics, Vol. 15, 2nd edn. Springer, Heidelberg (2002)

    Google Scholar 

  20. Briane M., Milton G.W., Nesi V.: Change of sign of the corrector’s determinant for homogenization in three-dimensional conductivity. Arch. Ration. Mech. Anal. 173(1), 133–150 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Caffarelli L.A., Souganidis P.E.: A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Comm. Pure Appl. Math. 61(1), 1–17 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Chu, C.-C., Graham, I.G., Hou, T.Y.: A New Multiscale Finite Element Method for High-Contrast Elliptic Interface Problems. Math. Comput., submitted 2009

  23. Donato P., Cioranescu D.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

  24. Conca C., Vanninathan M.: On uniform H 2-estimates in periodic homogenization. Proc. R. Soc. Edinb. Sect. A 131(3), 499–517 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Desbrun, M., Donaldson, R., Owhadi, H.: Discrete geometric structures in homogenization and inverse homogenization with application to eit. preprint arXiv:0904.2601, 2009

  26. Engquist W.E.B., Li X., Ren W., Vanden-Eijnden E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)

    MATH  MathSciNet  Google Scholar 

  27. Efendiev Y., Galvis J., Wu X.: Multiscale finite element and domain decomposition methods for high-contrast problems using local spectral basis functions, submitted 2009

  28. Efendiev Y., Ginting V., Hou T., Ewing R.: Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220(1), 155–174 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  29. Efendiev Y., Hou T.: Multiscale finite element methods for porous media flows and their applications. Appl. Numer. Math. 57(5–7), 577–596 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Engquist B., Souganidis P.E.: Asymptotic and numerical homogenization. Acta Numer. 17, 147–190 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  31. De Giorgi E.: Sulla convergenza di alcune successioni di integrali del tipo dell’aera. Rendi Conti di Mat. 8, 277–294 (1975)

    MATH  Google Scholar 

  32. De Giorgi, E.: New problems in Γ-convergence and G-convergence. In:Free Boundary Problems, Vol. II (Pavia, 1979), pp. 183–194. Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980

  33. Gloria A.: Analytical framework for the numerical homogenization of elliptic monotone operators and quasiconvex energies. SIAM MMS 5(3), 996–1043 (2006)

    MATH  MathSciNet  Google Scholar 

  34. Harbrecht H., Schneider R., Schwab C.: Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109(3), 385–414 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hou T.Y., Wu X.-H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. Jikov V.V., Kozlov S.M., Oleĭnik O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Heidelberg (1991)

    Google Scholar 

  37. Jikov V.V., Kozlov S.M., Oleĭnik O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994)

    Google Scholar 

  38. Kharevych, L., Mullen, P., Owhadi, H., Desbrun, M.: Numerical coarsening of inhomogeneous elastic materials. ACM Trans. Graphics (SIGGRAPH), 28(3), 2009

  39. Kikuchi, N., Oden, J.T.: Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM Studies in Applied Mathematics, Vol. 8. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988

  40. Kotiuga, P.R.: A rationale for pursuing eit and mreit in 3-d based on weyl asymptotics and problem conditioning. 12th Biennial Conference on Electromagnetic Field Computation, Miami, Florida, April 30–May 3, 2006

  41. Kozlov, S.M. (1979). The averaging of random operators. Mat. Sb. (N.S.) 109(151)(2), 188–202, 327 (1979)

    Google Scholar 

  42. Lassas M., Uhlmann G.: On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann. Sci. École Norm. Sup. (4) 34(5), 771–787 (2001)

    MATH  MathSciNet  Google Scholar 

  43. Lee J.M., Uhlmann G.: Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math. 42(8), 1097–1112 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  44. Leonardi S.: Weighted Miranda–Talenti inequality and applications to equations with discontinuous coefficients. Comment. Math. Univ. Carolin. 43(1), 43–59 (2002)

    MATH  MathSciNet  Google Scholar 

  45. Maugeri A., Palagachev D.K., Softova L.G.: Elliptic and Parabolic Equations with Discontinuous Coefficients, Mathematical Research, Vol. 109. Wiley-VCH, New York (2000)

    Book  Google Scholar 

  46. Melenk J.M.: On n-widths for elliptic problems. J. Math. Anal. Appl. 247(1), 272–289 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  47. Murat F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5(3), 489–507 (1978)

    MATH  MathSciNet  Google Scholar 

  48. Murat, F., Tartar, L.: H-convergence. Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger, 1978

  49. Netrusov Y., Safarov Y.: Weyl asymptotic formula for the Laplacian on domains with rough boundaries. Commun. Math. Phys. 253(2), 481–509 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  50. Nguetseng G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  51. Nolen J., Papanicolaou G., Pironneau O.: A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul. 7(1), 171–196 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  52. Owhadi H., Zhang L.: Errata to metric-based upscaling. Available at http://www.acm/caltech.edu/owhadi/, 2007

  53. Owhadi H., Zhang L.: Metric-based upscaling. Comm. Pure Appl. Math. 60(5), 675–723 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  54. Owhadi H., Zhang L.: Homogenization of parabolic equations with a continuum of space and time scales. SIAM J. Numer. Anal. 46(1), 1–36 (2007/2008)

    Article  MathSciNet  Google Scholar 

  55. Owhadi H., Zhang L.: Homogenization of the acoustic wave equation with a continuum of scales. Comp. Methods Appl. Mech. Eng. 198(2–4), 97–406 (2008) Arxiv math.NA/0604380

    MathSciNet  Google Scholar 

  56. Papanicolaou, G.C., Varadhan, S.R.S.: Diffusions with random coefficients. Statistics and Probability: Essays in Honor of C. R. Rao, North-Holland, Amsterdam, 547–552, 1982

  57. Pinkus A.: n-Width in Approximation Theory. Springer, New York (1985)

    Google Scholar 

  58. Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 571–597; errata, ibid. (3) 22, 673 (1968)

  59. Spagnolo, S.: Convergence in energy for elliptic operators. Numerical Solution of Partial Differential Equations, III. (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975), Academic Press, New York, 469–498, 1976

  60. Strouboulis T., Zhang L., Babuška I.: Assessment of the cost and accuracy of the generalized FEM. Int. J. Numer. Methods Eng. 69(2), 250–283 (2007)

    Article  MATH  Google Scholar 

  61. Todor R.A., Schwab C.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal. 27(2), 232–261 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  62. Weyl H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68(2), 220–269 (1910)

    Article  MATH  MathSciNet  Google Scholar 

  63. White, C.D., Horne, R.N.: Computing absolute transmissibility in the presence of finescale heterogeneity. SPE Symposium on Reservoir Simulation, 16011 (1987)

  64. Wu X.H., Efendiev Y., Hou T.Y.: Analysis of upscaling absolute permeability. Discrete Contin. Dyn. Syst. Ser. B 2(2), 185–204 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  65. Zhang, L., Berlyand, L., Federov, M., Owhadi, H.: Global energy matching method for atomistic to continuum modeling of self-assembling biopolymer aggregates. submitted, 2009

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, USA

    Leonid Berlyand

  2. Applied and Computational Mathematics, Control and Dynamical Systems, California Institute of Technology, MC 217-50, Pasadena, CA, 91125, USA

    Houman Owhadi

Authors
  1. Leonid Berlyand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Houman Owhadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Houman Owhadi.

Additional information

Communicated by C. Le Bris

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berlyand, L., Owhadi, H. Flux Norm Approach to Finite Dimensional Homogenization Approximations with Non-Separated Scales and High Contrast. Arch Rational Mech Anal 198, 677–721 (2010). https://doi.org/10.1007/s00205-010-0302-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-010-0302-1

Keywords

Search

Navigation