The Heterogeneous Multi-Scale Method for Homogenization Problems

  • Weinan E 
  • Björn Engquist 
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 44)


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.

Key words

multiscale problems homogenization heterogeneous multiscale method 


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  1. 1.
    G. Allaire, “Homogenization and two-scale convergence”. SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518.CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Abdulle, “Fourth order Chebychev methods with recurrence relations”, SIAM J. Sci. Comput., vol 23, pp. 2041–2054, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    A. Abdulle and W. E, “Finite difference HMM for homogenization problems”, J. Comput. Phys., 191, 18–39 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    I. Babuska, “Homogenization and its applications”, SYNSPADE 1975, B Hubbard ed. pp. 89–116.Google Scholar
  5. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 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.CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    A. Benssousan, J.L. Lions and G. Papanicolaou, “Asymptotic Analysis of Periodic Structures,” North-Holland (1978).Google Scholar
  8. 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.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 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. 10.
    L. T. Cheng and W. E, “HMM for Hamilton-Jacobi equations”, in preparation.Google Scholar
  11. 11.
    P. G. Ciarlet, “The Finite Element Methods for Elliptic Problems”, Amsterdam; New York: North-Holland Pub. Co., 1978.Google Scholar
  12. 12.
    L. Durlofsky, “Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media”, Water Resour. Res., 27, 699–708, 1991.CrossRefGoogle Scholar
  13. 13.
    W. E, “Homogenization of scalar conservation laws with oscillatory forcing terms”, SIAM J. Appl. Math., vol. 52, pp. 959–972, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    W. E, “Homogenization of linear and nonlinear transport equations”, Comm. Pure and Appl., vol. XLV, 301–326, 1992.MathSciNetGoogle Scholar
  15. 15.
    W. E and B. Engquist, “The heterogeneous multi-scale methods”, Comm. Math. Sci. 1, 87–133 (2003).zbMATHMathSciNetGoogle Scholar
  16. 16.
    W. E, B. Engquist and Z. Huang, Phys. Rev. B, 67(9), 092101 (2003).CrossRefGoogle Scholar
  17. 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. 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.CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    B. Engquist, “Computation of oscillatory solutions to hyperbolic differential equations”, Springer Lecture Notes in Mathematics, 1270, 10–22 (1987).zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    B. Engquist and O. Runborg, “Wavelet-based numerical homogenization with applications”, Lecture Notes in Computational Science and Engineering, T.J. Barth eds., Springer, 2002.Google Scholar
  21. 21.
    B. Engquist and R. Tsai, “HMM for stiff ODEs”, Math. Comp., in press.Google Scholar
  22. 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.zbMATHGoogle Scholar
  23. 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.CrossRefMathSciNetzbMATHGoogle Scholar
  24. 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. 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.CrossRefMathSciNetzbMATHGoogle Scholar
  26. 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).CrossRefzbMATHMathSciNetGoogle Scholar
  27. 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.MathSciNetzbMATHGoogle Scholar
  28. 28.
    R. LeVeque, “Numerical Methods for Conservation Laws,” Birkhäuser, 1990.Google Scholar
  29. 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. 30.
    P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, “Homogenization of Hamilton-Jacobi equations”, unpublished.Google Scholar
  31. 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.CrossRefzbMATHGoogle Scholar
  32. 32.
    P. B. Ming, “Analysis of multiscale methods”, in preparation.Google Scholar
  33. 33.
    P. B. Ming and X. Y. Yue, “Numerical Methods for Multiscale Elliptic Problems,” preprint, 2003.Google Scholar
  34. 34.
    S. Mueller, “Homogenization of nonconvex integral functionals and cellular materials”, Arch. Rat. Anal. Mech. vol. 99 (1987), 189–212.zbMATHGoogle Scholar
  35. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  36. 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. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  38. 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).CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    W. Ren and W. E, “HMM for the modeling of complex fluids and microfluidics”, J. Comput. Phys., to appear.Google Scholar
  40. 40.
    G. Samaey, D. Roose and I. G. Kevrekidis. “Combining the Gap-Tooth Scheme with Projective Integration: Patch Dynamics”, this volume.Google Scholar
  41. 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. 42.
    C. Schwab and A.-M. Matache, “Generalzied FEM for homogenization problems”, Lecture Notes in Computational Science and Engineering, T.J. Barth eds., Springer, 2002.Google Scholar
  43. 43.
    E. B. Tadmor, M. Ortiz and R. Phillips, “Quasicontinuum analysis of defects in crystals,” Phil. Mag., A73, 1529–1563 (1996).Google Scholar
  44. 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. 45.
    S. Torquato, “Random Heterogeneous Materials: Microstructure and Macroscopic Properties”, Springer-Verlag, 2001.Google Scholar
  46. 46.
    E. Vanden-Eijnden, “Numerical techniques for multiscale dynamical systems with stochastic effects”, Comm. Math. Sci., 1, 385–391 (2003).zbMATHMathSciNetGoogle Scholar
  47. 47.
    K. Xu and K. H. Prendergast, “Numerical Navier-Stokes solutions from gas kinetic theory,” J. Comput. Phys., 114, 9–17 (1994).CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Weinan E 
    • 1
    • 2
  • Björn Engquist 
    • 1
    • 3
  1. 1.Department of Mathematics and PACMPrinceton UniversityPrincetonUSA
  2. 2.School of MathematicsPeking UniversityBeijingChina
  3. 3.Department of MathematicsUniversity of Texas at AustinAustinUSA

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