Volume 48, Number 2, March-April 2014
Multiscale problems and techniques
Page(s) 517 - 552
Published online 11 March 2014
  1. A. Abdulle and M.J. Grote, Finite element heterogeneous multiscale method for the wave equation. Multiscale Model. Simul. 9 (2011) 766–792. [CrossRef] [Google Scholar]
  2. A. Abdulle and Ch. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces. Multiscale Model. Simul. 3 (2004) 195–220. [CrossRef] [Google Scholar]
  3. G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization. Multiscale Model. Simul. 4 (2005) 790–812. [CrossRef] [MathSciNet] [Google Scholar]
  4. T. Arbogast and K.J. Boyd. Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44 (2006) 1150–1171. [CrossRef] [Google Scholar]
  5. T. Arbogast, C.-S. Huang and S.-M. Yang, Improved accuracy for alternating-direction methods for parabolic equations based on regular and mixed finite elements. Math. Models Methods Appl. Sci. 17 (2007) 1279–1305. [CrossRef] [Google Scholar]
  6. S.N. Armstrong and P.E. Souganidis, Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments. J. Math. Pures Appl. 97 (2012) 460–504. [CrossRef] [Google Scholar]
  7. M. Atteia, Fonctions spline et noyaux reproduisants d’Aronszajn-Bergman. Rev. Française Informat. Recherche Opérationnelle 4 (1970) 31–43. [Google Scholar]
  8. 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]
  9. I. Babuška and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9 (2011) 373–406. [CrossRef] [MathSciNet] [Google Scholar]
  10. I. Babuška and J.E. Osborn, Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. [CrossRef] [MathSciNet] [Google Scholar]
  11. I. Babuška and J.E. Osborn, Can a finite element method perform arbitrarily badly? Math. Comput. 69 (2000) 443–462. [Google Scholar]
  12. G. Bal and W. Jing, Corrector theory for MSFEM and HMM in random media. Multiscale Model. Simul. 9 (2011) 1549–1587. [CrossRef] [Google Scholar]
  13. G. Ben Arous and H. Owhadi, Multiscale homogenization with bounded ratios and anomalous slow diffusion. Comm. Pure Appl. Math. 56 (2003) 80–113. [CrossRef] [MathSciNet] [Google Scholar]
  14. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structure. North Holland, Amsterdam (1978). [Google Scholar]
  15. L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast. Arch. Rational Mech. Anal. 198 (2010) 677–721. [Google Scholar]
  16. X. Blanc, C. Le Bris and P.-L. Lions, Une variante de la théorie de l’homogénéisation stochastique des opérateurs elliptiques. C. R. Math. Acad. Sci. Paris 343 (2006) 717–724. [Google Scholar]
  17. X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices. J. Math. Pures Appl. 88 (2007) 34–63. [CrossRef] [Google Scholar]
  18. A. Bourgeat and A. Piatnitski, Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal. 21 (1999) 303–315. [MathSciNet] [Google Scholar]
  19. L.V. Branets, S.S. Ghai, L.L. and X.-H. Wu, Challenges and technologies in reservoir modeling. Commun. Comput. Phys. (2009) 6 1–23. [Google Scholar]
  20. R.A. Brownlee, Error estimates for interpolation of rough and smooth functions using radial basis functions. Ph.D. thesis. University of Leicester (2004). [Google Scholar]
  21. L.A. Caffarelli and P.E. Souganidis, A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Comm. Pure Appl. Math. 61 (2008) 1–17. [CrossRef] [MathSciNet] [Google Scholar]
  22. C.-C. Chu, I.G. Graham and T.Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79 (2010) 1915–1955. [CrossRef] [Google Scholar]
  23. M. Desbrun, R. Donaldson and H. Owhadi. Modeling across scales: Discrete geometric structures in homogenization and inverse homogenization. Reviews of Nonlinear Dynamics and Complexity. Special issue on Multiscale Analysis and Nonlinear Dynamics (2012). [Google Scholar]
  24. J. Duchon, Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. Rev. Francaise Automat. Informat. Recherche Operationnelle Ser. RAIRO Anal. Numer. 10 (1976) 5–12. [Google Scholar]
  25. J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev spaces, in Constructive theory of functions of several variables, Proc. of Conf., Math. Res. Inst., Oberwolfach, 1976, in vol. 571. of Lect. Notes Math. Springer, Berlin (1977) 85–100. [Google Scholar]
  26. J. Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les Dm-splines. RAIRO Anal. Numér. (1978) 12 325–334. [MathSciNet] [Google Scholar]
  27. W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. [CrossRef] [MathSciNet] [Google Scholar]
  28. Y. Efendiev, J. Galvis and X. Wu, Multiscale finite element and domain decomposition methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937–955. [Google Scholar]
  29. Y. Efendiev, V. Ginting, T. Hou and R. Ewing, Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220 (2006) 155–174. [CrossRef] [Google Scholar]
  30. 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]
  31. Y. Efendiev and T.Y. Hou, Multiscale finite element methods, Theory and applications, in vol. 4, Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York (2009). [Google Scholar]
  32. I. Ekeland and R. Temam, Convex Analysis and Variational Problems, vol. 28 of Classics in Appl. Math. Society for Industrial and Applied Mathematics (1987). [Google Scholar]
  33. B. Engquist and P.E. Souganidis, Asymptotic and numerical homogenization. Acta Numerica 17 (2008) 147–190. [Google Scholar]
  34. B. Engquist, H. Holst and O. Runborg, Multi-scale methods for wave propagation in heterogeneous media. Commun. Math. Sci. 9 (2011) 33–56. [CrossRef] [Google Scholar]
  35. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, vol. 105. Ann. Math. Stud. Princeton University Press, Princeton, NJ (1983). [Google Scholar]
  36. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag (1983). [Google Scholar]
  37. E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell’aera. Rendi Conti di Mat. 8 (1975) 277–294. [Google Scholar]
  38. A. Gloria, Analytical framework for the numerical homogenization of elliptic monotone operators and quasiconvex energies. SIAM MMS 5 (2006) 996–1043. [CrossRef] [Google Scholar]
  39. A. Gloria, Reduction of the resonance error-Part 1: Approximation of homogenized coefficients. Math. Models Methods Appl. Sci. 21 (2011) 1601–1630. [CrossRef] [Google Scholar]
  40. A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 (2012) 1–28. [CrossRef] [Google Scholar]
  41. L. Grasedyck, I. Greff and S. Sauter, The al basis for the solution of elliptic problems in heterogeneous media. Multiscale Modeling and Simulation 10 (2012) 245–258. [Google Scholar]
  42. M. Grüter and K. Widman, The green function for uniformly elliptic equations. Manuscripta Math. 37 (1982) 303–342. [Google Scholar]
  43. R.L. Harder and R.N. Desmarais, Interpolation using surface splines. J. Aircr. 9 (1972) 189–191. [CrossRef] [Google Scholar]
  44. T.Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913–943. [CrossRef] [MathSciNet] [Google Scholar]
  45. T.Y. Hou and X.H. 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]
  46. V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer-Verlag (1991). [Google Scholar]
  47. E. Kosygina, F. Rezakhanlou and S.R.S. Varadhan, Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Comm. Pure Appl. Math. 59 (2006) 1489–1521. [CrossRef] [MathSciNet] [Google Scholar]
  48. O. Kounchev and H. Render, Polyharmonic splines on grids Z× aZn and their limits. Math. Comput. 74 (2005) 1831–1841. [CrossRef] [Google Scholar]
  49. S.M. Kozlov, The averaging of random operators. Mat. Sb. (N.S.) 109 (1979) 188–202, 327. [MathSciNet] [Google Scholar]
  50. P.-L. Lions and P.E. Souganidis, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Comm. Pure Appl. Math. 56 (2003) 1501–1524. [CrossRef] [MathSciNet] [Google Scholar]
  51. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. Approx. Theory Appl. 4 (1988) 77–89. [MathSciNet] [Google Scholar]
  52. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions. II. Math. Comput. 54 (1990) 211–230. [CrossRef] [Google Scholar]
  53. W.R. Madych and S.A. Nelson, Polyharmonic cardinal splines. J. Approx. Theory 60 (1990) 141–156. [CrossRef] [MathSciNet] [Google Scholar]
  54. W.R. Madych and S.A. Nelson, Polyharmonic cardinal splines: a minimization property. J. Approx. Theory 63 (1990) 303–320. [CrossRef] [MathSciNet] [Google Scholar]
  55. A. Malqvist and D. Peterseim, Localization of elliptic multiscale problems. Technical report arXiv:1110.0692 (2012). [Google Scholar]
  56. O.V. Matveev, Some methods for the reconstruction of functions of n variables defined on chaotic grids. Dokl. Akad. Nauk 326 (1992) 605–609. [Google Scholar]
  57. O.V. Matveev, Spline interpolation of functions of several variables and bases in Sobolev spaces. Trudy Mat. Inst. Steklov. 198 (1992) 125–152. [Google Scholar]
  58. O.V. Matveev, Interpolation of functions on chaotic grids. Dokl. Akad. Nauk 339 (1994) 594–597. [Google Scholar]
  59. O.V. Matveev, Methods for the approximate recovery of functions defined on chaotic grids. Izv. Ross. Akad. Nauk Ser. Mat. 60 111–156, 1996. [CrossRef] [MathSciNet] [Google Scholar]
  60. O.V. Matveev, On a method for the interpolation of functions on chaotic grids. Mat. Zametki 62 (1997) 404–417. [CrossRef] [MathSciNet] [Google Scholar]
  61. J.M. Melenk, On n-widths for elliptic problems. J. Math. Anal. Appl. 247 (2000) 272–289. [CrossRef] [Google Scholar]
  62. G.W. Milton, The theory of composites, vol. 6 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002). [Google Scholar]
  63. P. Ming and X. Yue, Numerical methods for multiscale elliptic problems. J. Comput. Phys. 214 (2006) 421–445. [CrossRef] [Google Scholar]
  64. R. Moser, Theory of partial differential equations. MA6000A. Lect. Notes (2012). Available on [Google Scholar]
  65. F. Murat and L. Tartar, H-convergence. Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger (1978). [Google Scholar]
  66. F.J. Narcowich, J.D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74 (2005) 743–763. [CrossRef] [Google Scholar]
  67. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. [CrossRef] [MathSciNet] [Google Scholar]
  68. 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]
  69. H. Owhadi and L. Zhang, Metric-based upscaling. Comm. Pure Appl. Math. 60 (2007) 675–723. [Google Scholar]
  70. H. Owhadi and L. Zhang. Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast. SIAM Multiscale Model. Simul. 9 (2011) 1373–1398. arXiv:1011.0986. [CrossRef] [Google Scholar]
  71. H. Owhadi, Anomalous slow diffusion from perpetual homogenization. Ann. Probab. 31 (2003) 1935–1969. [CrossRef] [Google Scholar]
  72. H. Owhadi, Averaging versus chaos in turbulent transport? Comm. Math. Phys. 247 (2004) 553–599. [CrossRef] [MathSciNet] [Google Scholar]
  73. G.C. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random fields, Vol. I, II (Esztergom (1979)), vol. 27. Colloq. Math. Soc. János Bolyai. North-Holland, Amsterdam (1981) 835–873. [Google Scholar]
  74. A. Pinkus, N-Widths in Approximation Theory. Springer-Verlag (1985). [Google Scholar]
  75. C. Rabut, B-splines Polyarmoniques Cardinales: Interpolation, Quasi-interpolation, filtrage. Thèse d’État. Université de Toulouse (1990). [Google Scholar]
  76. Ch. Rabut, Elementary m-harmonic cardinal B-splines. Numer. Algorithms 2 (1992) 39–61. [CrossRef] [MathSciNet] [Google Scholar]
  77. Ch. Rabut, High level m-harmonic cardinal B-splines. Numer. Algorithms 2 (1992) 63–84. [CrossRef] [MathSciNet] [Google Scholar]
  78. M. Rossini, Detecting discontinuities in two-dimensional signals sampled on a grid. JNAIAM J. Numer. Anal. Ind. Appl. Math. 4 (2009) 203–215. [MathSciNet] [Google Scholar]
  79. I.J. Schoenberg, Cardinal spline interpolation. Conference Board of the Mathematical Sciences Regional Conf. Ser. Appl. Math. No. 12. Society for Industrial and Applied Mathematics, Philadelphia, Pa., (1973). [Google Scholar]
  80. S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa 22 (1968) 571-597; errata, S. Spagnolo, Ann. Scuola Norm. Sup. Pisa 22 (1968) 673. [Google Scholar]
  81. G. Stampacchia, Le problème de dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. [CrossRef] [MathSciNet] [Google Scholar]
  82. G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus. Séminaire Jean Leray No. 3 (1963–1964). Numdam (1964). [Google Scholar]
  83. William Symes. Transfer of approximation and numerical homogenization of hyperbolic boundary value problems with a continuum of scales. TR12-20 Rice Tech Report (2012). [Google Scholar]
  84. J.L. Taylor, S. Kim and R.M. Brown, The green function for elliptic systems in two dimensions. arXiv:1205.1089 (2012). [Google Scholar]
  85. J. Vybiral, Widths of embeddings in function spaces. J. Complexity 24 (2008) 545–570. [CrossRef] [MathSciNet] [Google Scholar]
  86. C.D. White and R.N. Horne, Computing absolute transmissibility in the presence of finescale heterogeneity. SPE Symposium on Reservoir Simulation 16011 (1987). [Google Scholar]
  87. X.H. Wu, Y. Efendiev and T.Y. Hou, Analysis of upscaling absolute permeability. Discrete Contin. Dyn. Syst. Ser. B 2 (2002) 185–204. [Google Scholar]
  88. V.V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh. 27 (1986) 167–180. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you