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Numerical Approximations to Multiscale Solutions in Partial Differential Equations

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Frontiers in Numerical Analysis

Part of the book series: Universitext ((UTX))

Abstract

Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 105 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods such as homogenization, numerical samplings, multiscale finite element methods, variational multiscale methods, and wavelets based homogenization. Applications of these multiscale methods to transport through heterogeneous porous media and incompressible flows will be discussed. This paper is not intended to be a detailed survey and the discussion is limited by both the taste and expertise of the author.

Research was in part supported by a grant DMS-0073916 from the National Science Foundation

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References

  1. J. Aarnes and T. Y. Hou An Efficient Domain Decomposition Preconditioner for Multiscale Elliptic Problems with High Aspect Ratios, Acta Mathematicae Applicatae Sinica, 18 (2002), 63–76.

    Article  MathSciNet  MATH  Google Scholar 

  2. T. Arbogast, Numerical Subgrid Upscaling of Two-Phase Flow in Porous Media, in Numerical treatment of multiphase flows in porous media, Z. Chen et al., eds., Lecture Notes in Physics 552, Springer, Berlin, 2000, pp. 35–49.

    Chapter  Google Scholar 

  3. M. Avellaneda, T. Y. Hou and G. Papanicolaou, Finite Difference Approximations for Partial Differential Equations with Rapidly Oscillating Coefficients, Mathematical Modelling and Numerical Analysis, 25 (1991), 693–710.

    MathSciNet  MATH  Google Scholar 

  4. I. Babuska, G. Caloz, and 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.

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Babuska and E. Osborn, Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods, SIAM J. Numer. Anal., 20 (1983), 510–536.

    Article  MathSciNet  MATH  Google Scholar 

  6. I. Babuska and W. G. Szymezak, An Error Analysis for the Finite Element Method Applied to Convection-Diffusion Problems, Comput. Methods Appl. Math. Engrg, 31 (1982), 19–42.

    Article  MATH  Google Scholar 

  7. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Volume 5 of Studies in Mathematics and Its Applications, North-Holland Publ., 1978.

    Google Scholar 

  8. G. Beylkin, R. Coifman, and V. Rokhlin Fast Wavelet Transforms and Numerical Algorithm I Comm. Pure Appl. Math., 44 (1991), 141–183.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Bourgeat, Homogenized Behavior of Two-Phase Flows in Naturally Fractured Reservoirs with Uniform Fractures Distribution, Comp. Meth. Appl. Mech. Engrg, 47 (1984), 205–216.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Brewster and G. Beylk in, A Multiresolution Strategy for Numerical Homogenization, ACHA, 2 (1995), 327–349.

    MATH  Google Scholar 

  11. F. Brezzi and A. Russo, Choosing Bubbles for Advection-Diffusion Problems, Math. Models Methods Appl. Sci, 4 (1994), 571–587.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Brezzi, L. P. Franca, T. J. R. Hughes and A. Russo, b = ∫ g, Comput. Methods in Appl. Mech. and Engrg., 145 (1997), 329–339.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. E. Broadwell, Shock Structure in a Simple Discrete Velocity Gas, Phys. Fluids, 7 (1964), 1243–1247.

    Article  MATH  Google Scholar 

  14. T. Carleman, Problèms Maihématiques dans la Théorie Cinétique de Gaz, Publ. Sc. Inst. Mittag-Leffier, Uppsala, 1957.

    Google Scholar 

  15. H. Ceniceros and T. Y. Hou, An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions. J. Comput. Phys., 172 (2001), 609–639.

    Article  MATH  Google Scholar 

  16. Z. Chen and T. Y. Hou, A Mixed Finite Element Method for Elliptic Problems with Rapidly Oscillating Coefficients, to appear in Math. Comput..

    Google Scholar 

  17. A. J. Chorin, Vortex Models and Boundary Layer Instabilities, SIAM J. Sci. Statist. Comput., 1 (1980), 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. E. Cruz and A. Petera, A Parallel Monte-Carlo Finite Element Procedure for the Analysis of Multicomponent Random Media, Int. J. Numer. Methods Engrg, 38 (1995), 1087–1121.

    Article  MATH  Google Scholar 

  19. J. E. Dendy, J. M. Hyman, and J. D. Moulton, The Black Box Multigrid Numerical Homogenization Algorithm, J. Comput. Phys., 142 (1998), 80–108.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Daubechies Ten Lectures on Wavelets. SIAM Publications, 1991.

    Google Scholar 

  21. M. Dorobantu and B. Engquist, Wavelet-based Numerical Homogenization, SIAM J. Numer. Anal., 35 (1998), 540–559.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Douglas, Jr. and T.F. Russell, Numerical Methods for Convection-dominated Diffusion Problem Based on Combining the Method of Characteristics with Finite Element or Finite Difference Procedures, SIAM J. Numer. Anal. 19 (1982), 871–885.

    Article  MathSciNet  MATH  Google Scholar 

  23. L. J. Durlofsky, Numerical Calculation of Equivalent Grid Block Permeability Tensors for Heterogeneous Porous Media, Water Resour. Res., 27 (1991), 699–708.

    Article  Google Scholar 

  24. L.J. Durlofsky, R.C. Jones, and W.J. Milliken, A Nonuniform Coarsening Approach for the Scale-up of Displacement Processes in Heterogeneous Porous Media, Adv. Water Resources, 20 (1997), 335–347.

    Article  Google Scholar 

  25. B. B. Dykaar and P. K. Kitanidis, Determination of the Effective Hydraulic Conductivity for Heterogeneous Porous Media Using a Numerical Spectral Approach: 1. Method, Water Resour. Res., 28 (1992), 1155–1166.

    Article  Google Scholar 

  26. W. E and T. Y. Hou, Homogenization and Convergence of the Vortex Method for 2-D Euler Equations with Oscillatory Vorticity Fields, Comm. Pure and Appl. Math., 43 (1990), 821–855.

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. R. Efendiev, Multiscale Finite Element Method (MsFEM) and its Applications, Ph. D. Thesis, Applied Mathematics, Caltech, 1999.

    Google Scholar 

  28. Y. R. Efendiev, T. Y. Hou, and X. H. Wu, Convergence of A Non conforming Multiscale Finite Element Method, SIAM J. Numer. Anal., 37 (2000), 888–910.

    Article  MathSciNet  MATH  Google Scholar 

  29. Y. R. Efendiev, L. J. Durlofsky, S. H. Lee, Modeling of Subgrid Effects in Coarse-scale Simulation s of Transport in Heterogeneous Porous Media, WATER RESOUR RES, 36 (2000), 2031–2041.

    Article  Google Scholar 

  30. B. Engquist, Computation of Oscillatory Solutions to Partial Differential Equations, in Proc. Conference on Hyperbolic Partial Differential Equations, Carasso, Raviart, and Serre, eds, Lecture Notes in Mathematics No. 1270, Springer-Verlag, 10–22, 1987.

    Google Scholar 

  31. B. Engquist and T. Y. Hou, Particle Method Approximation of Oscillatory Solutions to Hyperbolic Differential Equations, SIAM J. Numer. Anal., 26 (1989), 289–319.

    Article  MathSciNet  MATH  Google Scholar 

  32. B. Engquist and T. Y. Hou, Computation of Oscillatory Solutions to Hyperbolic Equations Using Particle Methods, Lecture Notes in Mathematics No. 1360, Anderson and Greengard eds., Springer-Verlag, 68–82, 1988.

    Google Scholar 

  33. B. Engquist and H. O. Kreiss, Difference and Finite Element Methods for Hyperbolic Differential Equations, Comput. Methods Appl. Mech. Engrg., 17/18 (1979), 581–596.

    Article  Google Scholar 

  34. B. Engquist and E.D. Luo, Convergence of a Multigrid Method for Elliptic Equations with Highly Oscillatory Coefficients, SIAM J. Numer. Anal., 34 (1997), 2254–2273.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, New York, 2001.

    MATH  Google Scholar 

  36. J. Glimm, H. Kim, D. Sharp, and T. Wallstrom A Stochastic Analysis of the Scale Up Problem for Flow in Porous Media, Comput. Appl. Math., 17 (1998), 67–79.

    MathSciNet  MATH  Google Scholar 

  37. T. Y. Hou, Homogenization for Semilinear Hyperbolic Systems with Oscillatory Data, Comm. Pure and Appl. Math., 41 (1988), 471–495.

    Article  MathSciNet  MATH  Google Scholar 

  38. 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.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Y. Hou and X. H. Wu, A Multiscale Finite Element Method for PDEs with Oscillatory Coefficients, Proceedings of 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems, Jan 24–26, 1997, Notes on Numerical Fluid Mechanics, Vol. 70, ed. by W. Hackbusch and G. Wittum, Vieweg-Verlag, 58–69, 1999.

    Google Scholar 

  40. 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.

    Article  MathSciNet  MATH  Google Scholar 

  41. T. Y. Hou and D.-P. Yang, Convection of Microsiructure in Two and Three Dimensional Incompressible Euler Equations, in preparation, 2002.

    Google Scholar 

  42. T. J. R. Hughes, Multi scale Phenomena: Green’s Functions, the Dirichlet-to-Neumann Formulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods, Comput. Methods Appl. Mech Engrg., 127 (1995), 387–401.

    Article  MathSciNet  MATH  Google Scholar 

  43. T. J. R. Hughes, G. R. Feijoo, L. Mazzei, J.-B. Quincy, The Variational Multiscale Method — A Paradigm for Computational Mechanics, Comput. Methods Appl, Mech Engrg., 166 (1998), 3–24.

    Article  MathSciNet  MATH  Google Scholar 

  44. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994, Translated from Russian.

    Google Scholar 

  45. S. Knapek, Matrix-Dependent Multigrid-Homogenization for Diffusion Problems, in the Proceedings of the Copper Mountain Conference on Iterative Methods, edited by T. Manteuffal and S. McCormick, volume I, SIAM Special Interest Group on Linear Algebra, Cray Research, 1996.

    Google Scholar 

  46. P. Langlo and M.S. Espedal, Macrodispersion for Two-phase, Immiscible Flow in Porous Media, Adv. Water Resources 17 (1994), 297–316.

    Article  Google Scholar 

  47. A. M. Matache, I. Babuska, and C. Schwab, Generalized p-FEM in Homogenization, Numer. Math. 86 (2000), 319–375.

    Article  MathSciNet  MATH  Google Scholar 

  48. A. M. Matache and C. Schwab, Homogenization via p-FEM for Problems with Microstructure, Appl. Numer. Math. 33 (2000), 43–59.

    Article  MathSciNet  MATH  Google Scholar 

  49. J. F. McCarthy, Comparison of Fast Algorithms for Estimating Large-Scale Permeabilities of Heterogeneous Media, Transport in Porous Media, 19 (1995), 123–137.

    Article  Google Scholar 

  50. D. W. McLaughlin, G. C. Papanicolaou, and L. Tartar, Weak Limits of Semilinear Hyperbolic Systems with Oscillating Data, Lecture Notes in Physics 230 (1985), 277–289, Springer-Verlag, Berlin, New York.

    Google Scholar 

  51. D. W. McLaughlin, G. C. Papanicolaou, and O. Pironneau, Convection of Microsiructure and Related Problems, SIAM J. Applied Math, 45 (1985), 780–797.

    Article  MathSciNet  MATH  Google Scholar 

  52. S. Moskow and M. Vogelius, First Order Corrections to the Homogenized Eigenvalues of a Periodie Composite Medium: A Convergenee Proof, Proc. Roy. Soc. Edinburgh, A, 127 (1997), 1263–1299.

    Article  MathSciNet  MATH  Google Scholar 

  53. F. Murat, Compacité par Compensation, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489–507.

    MathSciNet  MATH  Google Scholar 

  54. F. Murat, Compacité par compensation II’ Proceedings of t he International Meeting on Recent Methods in Nonlinear Analysis, Rome, May 8–12, 1978, ed. by E. De Giorgi, E. Magenes and U. Mosco, Pitagora Editrice, Bologna, 245–256, 1979.

    Google Scholar 

  55. H. Neiderriter, Quasi-Monte Carlo Methods and Pseudo-Random Numbers, Bull. Amer. Math. Soc., 84 (1978), 957–1041.

    Article  MathSciNet  Google Scholar 

  56. O. Pironneau, On the Transport-diffusion Algorithm and its Application to the Navier-Stokes Equations, Numer. Math. 38 (1982), 309–332.

    Article  MathSciNet  MATH  Google Scholar 

  57. G. Sangalli, Capturing Small Scales in Elliptic Problems Using a Residual-Free Bubbles Finite Element Method, to appear in Multiscale Modeling and Simulation.

    Google Scholar 

  58. L. Tartar, Compensated Compactness and Application s to P.D.E., Nonlinear Analysis and Mechanics, Heriot-Watt Symposium, Vol. IV, ed. by R. J. Knops, Research Notes in Mathematics 39, Pitman, Boston, 136–212, 1979.

    Google Scholar 

  59. L. Tartar, Solutions oscillantes des équaiions de Carleman, Seminaire Goulaouic-Meyer-Schwartz (1980–1981), exp. XII. Ecole Polytechnique (Palaiseau), 1981.

    Google Scholar 

  60. L. Tartar, Nonlocal Effects Induced by Homogenization, in PDE and Calculus of Variations, ed by F. Culumbini, et al, Birkhäuser, Boston, 925–938, 1989.

    Google Scholar 

  61. X.H. Wu, Y. Efendiev, and T. Y. Hou, Analysis of Upscaling Absolute Permeability, Discrete and Continuous Dynamical Systems, Series B, 2 (2002), 185–204.

    Article  MathSciNet  MATH  Google Scholar 

  62. P. M. De Zeeuw, Matrix-dependent Prolongation and Restrictions in a Blackbox Multigrid Solver, J. Comput. Applied Math, 33 (1990), 1–27.

    Article  MATH  Google Scholar 

  63. S. Verdiere and M.H. Vignal, Numerical and Theoretical Study of a Dual Mesh Method Using Finite Volume Schemes for Two-phase Flow Problems in Porous Media, Numer. Math. 80 (1998), 601–639.

    Article  MathSciNet  MATH  Google Scholar 

  64. T. C. Wallstrom, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, Effective Flux Boundary Conditions for Upscaling Porous Media Equations, Transport in Porous Media, 46 (2002), 139–153.

    Article  MathSciNet  Google Scholar 

  65. T. C. Wallstrom, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, Application of Effective Flux Boundary Conditions to Two-phase Upscaling in Porous Media, Transport in Porous Media, 46 (2002), 155–178.

    Article  MathSciNet  Google Scholar 

  66. T. C. Wallstrom, S. L. Hou, M. A. Christie, L. J. Durlofsky, and D. H. Sharp, Accurate Scale Up of Two Phase Flow Using Renormalization and Nonuniform Coarsening, Comput. Geosci, 3 (1999), 69–87.

    Article  MATH  Google Scholar 

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  1. Applied Mathematics, 217-50, Caltech, Pasadena, CA, 91125, USA

    Thomas Y. Hou

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  1. Department of Mathematical Sciences, University of Durham, South Road, DH1 3LE, Durham, UK

    James F. Blowey  & Alan W. Craig  & 

  2. Department of Mathematics, University of Manchester, Oxford Road, M13 9PL, Manchester, UK

    Tony Shardlow

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Hou, T.Y. (2003). Numerical Approximations to Multiscale Solutions in Partial Differential Equations. In: Blowey, J.F., Craig, A.W., Shardlow, T. (eds) Frontiers in Numerical Analysis. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55692-0_6

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  • DOI: https://doi.org/10.1007/978-3-642-55692-0_6

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