Skip to main content
SpringerLink
Log in
Book cover

Multiscale Methods in Science and Engineering pp 89–110Cite as

The Heterogeneous Multi-Scale Method for Homogenization Problems

  • Conference paper

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 44))

Summary

The heterogeneous multi-scale method, a general framework for efficient numerical modeling of problems with multi-scales [15], is applied to a large variety of homogenization problems. These problems can be either linear or nonlinear, periodic or non-periodic, stationary or dynamic. Stability and accuracy issues are analyzed along the lines of the general principles outlined in [15]. Strategies for obtaining the microstructural information are discussed.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Allaire, “Homogenization and two-scale convergence”. SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Abdulle, “Fourth order Chebychev methods with recurrence relations”, SIAM J. Sci. Comput., vol 23, pp. 2041–2054, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Abdulle and W. E, “Finite difference HMM for homogenization problems”, J. Comput. Phys., 191, 18–39 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Babuska, “Homogenization and its applications”, SYNSPADE 1975, B Hubbard ed. pp. 89–116.

    Google Scholar 

  5. I. Babuska, “Solution of interface problems by homogenization”, I: SIAM J. Math. Anal., 7 (1976), no. 5, pp. 603–634. II: SIAM J. Math. Anal., 7 (1976), no. 5, pp. 635–645. III: SIAM J. Math. Anal., 8 (1977), no. 6, pp. 923–937.

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Babuska, G. Caloz and J. Osborn, “Special Finite Element Methods for a Class of Second Order Elliptic Problems with Rough Coefficients”, SIAM J. Numer. Anal., vol. 31, pp. 945–981, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Benssousan, J.L. Lions and G. Papanicolaou, “Asymptotic Analysis of Periodic Structures,” North-Holland (1978).

    Google Scholar 

  8. F. Brezzi, D. Marini, and E. Suli, “Residual-free bubbles for advection-diffusion problems: the general error analysis.” Numerische Mathematik. 85 (2000) 1, 31–47.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. T. Cheng and W. E, “HMM for interface dynamics”, in Contemporary Mathematics: A special volume in honor of Stan Osher, S. Y. Cheng, C. W. Shu and T. Tang eds.

    Google Scholar 

  10. L. T. Cheng and W. E, “HMM for Hamilton-Jacobi equations”, in preparation.

    Google Scholar 

  11. P. G. Ciarlet, “The Finite Element Methods for Elliptic Problems”, Amsterdam; New York: North-Holland Pub. Co., 1978.

    Google Scholar 

  12. L. Durlofsky, “Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media”, Water Resour. Res., 27, 699–708, 1991.

    Article  Google Scholar 

  13. W. E, “Homogenization of scalar conservation laws with oscillatory forcing terms”, SIAM J. Appl. Math., vol. 52, pp. 959–972, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  14. W. E, “Homogenization of linear and nonlinear transport equations”, Comm. Pure and Appl., vol. XLV, 301–326, 1992.

    MathSciNet  Google Scholar 

  15. W. E and B. Engquist, “The heterogeneous multi-scale methods”, Comm. Math. Sci. 1, 87–133 (2003).

    MATH  MathSciNet  Google Scholar 

  16. W. E, B. Engquist and Z. Huang, Phys. Rev. B, 67(9), 092101 (2003).

    Article  Google Scholar 

  17. W. E, D. Liu and E. Vanden-Eijnden, “Analysis of Multiscale Methods for Stochastic Differential Equations,” submitted to Comm. Pure Appl. Math., 2003.

    Google Scholar 

  18. W. E, P. Ming and P. W. Zhang, “Analysis of the heterogeneous multi-scale method for elliptic homogenization problems”, J. Amer. Math. Soc. vol 18, pp 121–156, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Engquist, “Computation of oscillatory solutions to hyperbolic differential equations”, Springer Lecture Notes in Mathematics, 1270, 10–22 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Engquist and O. Runborg, “Wavelet-based numerical homogenization with applications”, Lecture Notes in Computational Science and Engineering, T.J. Barth et.al eds., Springer, 2002.

    Google Scholar 

  21. B. Engquist and R. Tsai, “HMM for stiff ODEs”, Math. Comp., in press.

    Google Scholar 

  22. L. C. Evans, “The perturbed test function method for viscosity solutions of nonlinear PDE”, Proc. Royal Soc. Edinburgh Sect. A, 111 (1989), pp. 359–375.

    MATH  Google Scholar 

  23. C.W. Gear and I. G. Kevrekidis, “Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum,” SIAM J. Sci. Comp., vol 24, pp 1091–1106, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Gulliou and B. Lago, Domaine de stabilité associé aux formules d'intégration numérique d'équations différentielles, à pas séparés et à pas liés. ler Congr. Assoc. Fran. Calcul, AFCAL, Grenoble, pp. 43–56, Sept. 1960.

    Google Scholar 

  25. T. Hou and X. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media”, J. Comput. Phys., 134(1), 169–189, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. J. R. Hughes, “Multiscale phenomena: Green's functions, the Dirichlet to Neumann formulation, subgrid, scale models, bubbles and the origin of stabilized methods”, Comput. Methods Appl. Mech. Engrg., 127, 387–401 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  27. V. I. Lebedev and S. I. Finogenov, “Explicit methods of second order for the solution of stiff systems of ordinary differential equations”, Zh. Vychisl. Mat. Mat Fiziki, vol. 16, No. 4, pp. 895–910, 1976.

    MathSciNet  MATH  Google Scholar 

  28. R. LeVeque, “Numerical Methods for Conservation Laws,” Birkhäuser, 1990.

    Google Scholar 

  29. X. T. Li and W. E, “Multiscale modeling of the dynamics of solids at finite temperature”, submitted to J. Mech. Phys. Solids.

    Google Scholar 

  30. P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, “Homogenization of Hamilton-Jacobi equations”, unpublished.

    Google Scholar 

  31. W. K. Liu, Y. F. Zhang and M. R. Ramirez, “Multiple Scale Finite Element Methods”, International Journal for Numerical Methods in Engineering, vol. 32, pp. 969–990, 1991.

    Article  MATH  Google Scholar 

  32. P. B. Ming, “Analysis of multiscale methods”, in preparation.

    Google Scholar 

  33. P. B. Ming and X. Y. Yue, “Numerical Methods for Multiscale Elliptic Problems,” preprint, 2003.

    Google Scholar 

  34. S. Mueller, “Homogenization of nonconvex integral functionals and cellular materials”, Arch. Rat. Anal. Mech. vol. 99 (1987), 189–212.

    MATH  Google Scholar 

  35. G. Nguetseng, “A general convergence result for a functional related to the theory of homogenization”. SIAM J. Math. Anal. 20 (1989), no. 3, 608–623.

    Article  MATH  MathSciNet  Google Scholar 

  36. M. A. Novotny, “A tutorial on advanced dynamic Monte Carlo methods for systems with discrete state spaces”, Ann. Rev. Comput. Phys., pp. 153–210 (2001).

    Google Scholar 

  37. J. T. Oden and K. S. Vemaganti, “Estimation of local modelling error and global-oriented adaptive modeling of heterogeneous materials: error estimates and adaptive algorithms”, J. Comput. Phys., 164, 22–47 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  38. S. Osher and C. W. Shu, “High order essentially non-oscillatory schemes for Hamilton-Jacobi equations”, SIAM J. Numer. Anal., vol. 28, pp. 907–922 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  39. W. Ren and W. E, “HMM for the modeling of complex fluids and microfluidics”, J. Comput. Phys., to appear.

    Google Scholar 

  40. G. Samaey, D. Roose and I. G. Kevrekidis. “Combining the Gap-Tooth Scheme with Projective Integration: Patch Dynamics”, this volume.

    Google Scholar 

  41. C. Schwab, “Two-scale FEM for homogenization problems”, Proceedings of the Conference “Mathematical Modelling and Numerical Simulation in Continuum Mechanics”, Yamaguchi, Japan, I. Babuska, P. G. Ciarlet and T. Myoshi (Eds.), Lecture Notes in Computational Science and Engineering, Springer Verlag 2002.

    Google Scholar 

  42. C. Schwab and A.-M. Matache, “Generalzied FEM for homogenization problems”, Lecture Notes in Computational Science and Engineering, T.J. Barth et.al eds., Springer, 2002.

    Google Scholar 

  43. E. B. Tadmor, M. Ortiz and R. Phillips, “Quasicontinuum analysis of defects in crystals,” Phil. Mag., A73, 1529–1563 (1996).

    Google Scholar 

  44. L. Tartar, “Solutions oscillantes des équations de Carleman”. Goulaouic-Meyer-Schwartz Seminar, 1980–-1981, Exp. No. XII, 15 pp., École Polytech., Palaiseau, 1981.

    Google Scholar 

  45. S. Torquato, “Random Heterogeneous Materials: Microstructure and Macroscopic Properties”, Springer-Verlag, 2001.

    Google Scholar 

  46. E. Vanden-Eijnden, “Numerical techniques for multiscale dynamical systems with stochastic effects”, Comm. Math. Sci., 1, 385–391 (2003).

    MATH  MathSciNet  Google Scholar 

  47. K. Xu and K. H. Prendergast, “Numerical Navier-Stokes solutions from gas kinetic theory,” J. Comput. Phys., 114, 9–17 (1994).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and PACM, Princeton University, Princeton, NJ, 08544, USA

    E Weinan & Engquist Björn

  2. School of Mathematics, Peking University, Beijing, 100871, China

    E Weinan

  3. Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, Texas, 78712, USA

    Engquist Björn

Authors
  1. E Weinan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Engquist Björn
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Numerical Analysis and Computer Science, Royal Institute of Technology, SE-10044, Stockholm, Sweden

    Björn Engquist  & Olof Runborg  & 

  2. Department of Information Technology, Uppsala University, Box 337, SE-75105, Uppsala, Sweden

    Per Lötstedt

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Weinan, E., Björn, E. (2005). The Heterogeneous Multi-Scale Method for Homogenization Problems. In: Engquist, B., Runborg, O., Lötstedt, P. (eds) Multiscale Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26444-2_4

Download citation

Publish with us

Policies and ethics

Search

Navigation