Volume 48, Number 2, March-April 2014
Multiscale problems and techniques
Page(s) 493 - 515
Published online 11 March 2014
  1. T. Arbogast and K.J. Boyd, Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44 (2006) 1150–1171. [CrossRef] [Google Scholar]
  2. I. Babuška, Homogenization and Its Application. Mathematical and Computational Problems. SYNSPADE 1975, Numer. Solution Part. Differ. Eqs. lll, edited by B. Hubbard. Academic Press (1976) 89–116. [Google Scholar]
  3. I. Babuška, B. Anderson, P. Smith and K. Levin, Damage analysis of fiber composites, Part I Statistical analysis on fiber scale. Comput. Methods Appl. Mech Engrg. 172 (1999) 27–77. [CrossRef] [MathSciNet] [Google Scholar]
  4. I. Babuška, U. Banerjee and J. Osborn, Generalized Finite Element Methods-Main Ideas, Results and Perspective. Int. J. Comput. Methods 1 (2004) 67–103. [CrossRef] [Google Scholar]
  5. I. Babuška and U. Banerjee, Stable Generalized Finiter Element Methods (SGFEM). Comput. Meth. Appl. Mech. Eng. 201-204 (2012) 91–111. [CrossRef] [Google Scholar]
  6. I. Babuška, G. Caloz and J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945–981. [Google Scholar]
  7. I. Babuška and R. Lipton, Optimal local approximation spaces for Generalized Finite Element Methods with application to multiscale problems. Multiscale Model. Simul., SIAM 9 (2011) 373–406. [CrossRef] [MathSciNet] [Google Scholar]
  8. I. Babuška and R. Lipton, L2 global to local projection: an approach to multiscale analysis. M3AS 21 (2011) 2211–2226. [Google Scholar]
  9. I. Babuška and J. Melenk, The Partition of Unity Finite Element Method. Internat. J. Numer. Methods Engrg. 40 (1997) 727–758. [Google Scholar]
  10. I. Babuška, J, E. Osborn, Generalized finite element methods:Their performance and their relation to the mixed methods. SIAM, J. Numer. Anal. 20 (1983) 510–536. [Google Scholar]
  11. I. Babuška and J.E. Osborn, Eigenvalue Problems. Handbook of Numerical Analysis, Finite Element Methods (Part 1), Vol. II, edited by P.G. Ciarlet and J.L. Lions. Elsevier Science Publishers, Amsterdam (1991). [Google Scholar]
  12. N.S. Bakhvalov and G. Panasenko, Homogenization Processes in Periodic Media. Nauka, Moscow (1984). [Google Scholar]
  13. R.M. Barrer, Diffusion and permeation in heterogenous media. Diffusion in Polymers, edited by J. Crank, G.S. Park. Academic Press (1968). [Google Scholar]
  14. M. Bebendorf and W. Hackbusch, Existence of ℋ-matrix approximants to the inverse FE-matrix of elliptic operators with L coefficients. Numer. Math. 95 (2003) 1–28. [CrossRef] [MathSciNet] [Google Scholar]
  15. L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with nonseparated length scales and high contrast. Arch. Rat. Mech. Anal. 198 (2010) 177–221. [Google Scholar]
  16. A. Besounssan, J.L. Lions and G.C. Papanicolau, Asymptotic Analysis for Periodic Structures. North Holland Pub., Amsterdam (1978). [Google Scholar]
  17. T. Burchuladze and R. Rukhadze, Asymptotic distribution of eigenfunctions and eigenvalues of the basic boundary-contact oscillation problems of the classical theory of elasticity. Georgian Math. J. 6 (1999) 107–126. [CrossRef] [MathSciNet] [Google Scholar]
  18. C.C. Chams and G.P. Sendeckij, Critique on theories predicting thermoelastic properties of fibrous composites. J. Comput. Mat. 2 (1968) 332–358. [CrossRef] [Google Scholar]
  19. W. E, B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. [CrossRef] [MathSciNet] [Google Scholar]
  20. Weinan E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc. 18 (2005) 121–156. [CrossRef] [MathSciNet] [Google Scholar]
  21. Y. Efendiev and T.Y. Hou, Multiscale Finite Element Methods. Springer (2009). [Google Scholar]
  22. Y.R. Efendiev, T.Y. Hou and X.H. Wu, Convergence of a nonconforming mutiscale finite element method. SIAM J. Numer. Anal. 37 (2000) 888–910. [CrossRef] [MathSciNet] [Google Scholar]
  23. Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications. Appl. Numer. Math. 57 (2007) 577–596. [CrossRef] [Google Scholar]
  24. B. Engquist and P.E. Souganidis, Asymptotic and numerical homogenization. Acta Numer. 17 (2008) 147–190. [CrossRef] [MathSciNet] [Google Scholar]
  25. J. Fish and Z. Huan, Multiscale enrichment based on partition unity. Int. J. Num. Mech. Eng. 62 (2005) 1341–1359. [CrossRef] [Google Scholar]
  26. S.K. Garg, V. Svalbonas and G.A. Gurtman, Analysis of Structural Composite Materials. Marcel Dekker, New York (1973). [Google Scholar]
  27. L. Grasedyck, I. Greff and S. Sauter, The AL basis for the solution of elliptic problems in heterogeneous media. Multiscale Model. Simul. 10 (2012) 245–258. [CrossRef] [Google Scholar]
  28. L.J. Gurtman and R.H. Krock, Composite Materials, in vol II of Mechanics of composite materials, edited by G.P. Sendeckyj. Academic Press (1974). [Google Scholar]
  29. Z. Hashin, Theory of Fiber reinforced materials, NASA Report CR-1974 (1972) 1–704. [Google Scholar]
  30. T.Y. Hou and Xiao-Hui Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. [CrossRef] [MathSciNet] [Google Scholar]
  31. T.Y. Hou, Xiao-Hui Wu and Yu Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation. Commun. Math. Sci. 2 (2004) 185–205. [CrossRef] [Google Scholar]
  32. T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulaion, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401. [CrossRef] [MathSciNet] [Google Scholar]
  33. T.J.R. Hughes, G.R. Feijoo, L. Mazzei and J.B. Quincy, The variational multiscale method. A Paradigm for computational mechanics. Comput. Meth. Appl. Mech. Eng. 166 (1998) 3–24. [Google Scholar]
  34. K. Lichtenecker, Die Electrizitatskonstante naturlicher und kustlicher Mischkorper. Phys. Zeitschr. XXVII (1926) 115–158. [Google Scholar]
  35. A. Malquist, Multiscale methods for elliptic problems. Multiscale Model. Simul. 9 (2011) 1064–1086. [CrossRef] [Google Scholar]
  36. G.F. Masotti, Discussione analitica sul influenze che L’azione di mezo dialettrico hu sulla distribuziione dell’ electtricita alla superficie di pin corpi ellecttici diseminati in esso. Mem.Di Math. et di Fisica in Modena 24 (1850) 49. [Google Scholar]
  37. G. Maxwell, Trestise on Electricity and Magnetisum, vol. 1. Oxford Univ. Press (1873) 62. [Google Scholar]
  38. J. Melenk and I. Babuška, The Partion of Unity Method Basic Theory and Applications, Comput. Meth. Appl. Mech. Eng. 139 (1996) 289–314. [Google Scholar]
  39. J.M. Melenk, On n-widths for elliptic problems. J. Math. Anal. Appl. 247 (2000) 272–289. [CrossRef] [Google Scholar]
  40. G.W. Milton. The Theory of Composites. Cambridge University Press, Cambridge (2002). [Google Scholar]
  41. F. Murat, H-convergence, Séminaire d’Analyse Fonctionelle et Numérique de l’Université d’Alger, mimeographed notes, 1978. L. Tartar Cours Peccot, College de France (1977). Translated into English as F. Murat L. Tartar, H- convergence, in Topics in the Mathematical Modeling of Composite Materials, Progress in Nonlinear Differential Equations and their Applications, in vol. 31, edited by A.V. Cherkaev, R.V. Kohn. Birkhäuser, Boston (1997) 21–43. [Google Scholar]
  42. J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul. 7 (2008) 171–196. [CrossRef] [Google Scholar]
  43. H. Owhadi and L. Zhang, Metric-based upscaling. Commun. Pure Appl. Math. 60 (2007) 675–723. [CrossRef] [Google Scholar]
  44. H. Owhadi and L. Zhang, Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast. Multiscale Model. Simul. 9 (2011) 1373–1398. [CrossRef] [Google Scholar]
  45. A. Pinkus, n-Widths in Approximation Theory. Springer-Verlag, Berlin, Heidelberg, New York 7 (1985). [Google Scholar]
  46. S.D. Poisson, Second mem. sur la theorie de magnetism, Mem. de L Acad. de France (1822) 5. [Google Scholar]
  47. J.W. Rayleigh, On the influence of obstacles in rectangular order upon the properties of the medium. Philos. Mag. 50 (1892) 481. [CrossRef] [Google Scholar]
  48. E. Sanchez-Palencia. Non-Homogeneous Media and Vibration Theory, in vol. 127 of Lecture Notes in Physics. Springer-Verlag (1980). [Google Scholar]
  49. S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Ann Scu. Norm. Pisa 21 (1967) 657–699. [Google Scholar]
  50. S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa 22 (1968) 517–597. [Google Scholar]
  51. S. Spagnolo, Convergence in Energy for Elliptic Operators, edited by B. Hubbard. Numer. Solutions Partial Differ Eqs. III, (Synspade 1975, College Park, Maryland 1975). Academic Press, New York (1975). [Google Scholar]
  52. W. Streider and R. Aris, Variational Methods Applied to Problems of Diffusion and Reaction, Springer Tracts in Natural Philosophy. Springer-Verlag (1973). [Google Scholar]
  53. T. Strouboulis, L. Zhang and I Babuška, Generalized finite element method using mesh-based handbooks application to problem in domains with many voids. Comput. Methods Appl. Mechanics Engrg. 192 (2003) 3109–3161. [CrossRef] [Google Scholar]
  54. T. Strouboulis, I. Babuška and K. Copps, The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Engrg. 181 (2001) 43–69. [Google Scholar]
  55. T. Strouboulis, L. Zhang and I. Babuška, p-version of generalized FEM using mesh based handbooks with applications to multiscale problems. Int. J. Num. Meth. Engrg. 60 (2004) 1639–1672. [CrossRef] [Google Scholar]
  56. S. Torquato, Random Heterogeneous Materials, Microstructure and Macroscopic Properties. Springer, New York (2002). [Google Scholar]

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