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Asymptotic and numerical homogenization

  • B. Engquist (a1) and P. E. Souganidis (a1)

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Homogenization is an important mathematical framework for developing effective models of differential equations with oscillations. We include in the presentation techniques for deriving effective equations, a brief discussion on analysis of related limit processes and numerical methods that are based on homogenization principles. We concentrate on first- and second-order partial differential equations and present results concerning both periodic and random media for linear as well as nonlinear problems. In the numerical sections, we comment on computations of multi-scale problems in general and then focus on projection-based numerical homogenization and the heterogeneous multi-scale method.

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Abdulle, A. and W. E, (2003), ‘Finite difference heterogeneous multi-scale method for homogenization problems’, J. Comput. Phys. 191, 1839.
Allaire, G. (1992), ‘Homogenization and two-scale convergence’, SIAM J. Math. Anal. 23, 14821518.
Allaire, G., Braides, A., Buttazzo, G., Defranceschi, A. and Gibiansky, L. (1993), School on Homogenization. SISSA Ref. 140/73/M.
Andersson, U., Engquist, B., Ledfelt, G. and Runborg, O. (1999), ‘A contribution to wavelet-based subgrid modeling’, Appl. Comput. Harmon. Anal. 7, 151164.
Babuška, I. (1976), Homogenization and its applications: Mathematical and computational problems. In Numerical Solution of Partial Differential Equations III, Academic Press, pp. 89116.
Babuška, I., Caloz, G. and Osborn, E. (1994), ‘Special finite element methods for a class of second order elliptic problems with rough coefficients’, SIAM J. Numer. Anal. 31, 945981.
Barles, G. and Perthame, B. (1988), ‘Exit time problems in optimal control and vanishing viscosity solutions of Hamilton–Jacobi equations’, SIAM J. Control Optim. 26, 11331148.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978), Asymptotic Analysis for Periodic Structures, North-Holland.
Beylkin, G. and Brewster, M. (1995), ‘A multiresolution strategy for numerical homogenization’, Appl. Comput. Harmon. Anal. 2, 327349.
Beylkin, G., Coifman, R. and Rokhlin, V. (1991), ‘Fast wavelet transforms and numerical algorithms I’, Comm. Pure Appl. Math. 44, 141183.
Caffarelli, L. A. and Souganidis, P. E. (2007), Error estimates for the homogenization of uniformly elliptic pde in strongly mixing random media. Preprint.
Caffarelli, L. A. and Souganidis, P. E. (2008), ‘A rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic PDE’, Comm. Pure Appl. Math. 61, 117.
Caffarelli, L. A., Souganidis, P. E. and Wang, L. (2005), ‘Stochastic homogenization for fully nonlinear, second-order partial differential equations’, Comm. Pure Appl. Math. 30, 319361.
Capuzzo-Dolcetta, I. and Ishii, H. (2001), ‘On the rate of convergence in homogenization of Hamilton–Jacobi equations’, Indiana U. Math. J. 50, 110129.
Cardaliaguet, P., Lions, P.-L. and Souganidis, P. E. (2008), ‘A discussion about the homogenization of moving interfaces’, J. Mathématique Pure et Appliqué, to appear.
Cioranescu, D. and Donato, P. (2000), An Introduction to Homogenization, Oxford University Press.
Craciun, B. and Bhattachayra, K. (2003), ‘Homogenization of a Hamilton–Jacobi equation associated with the geometric motion of an interface’, Proc. Roy. Soc. Edinburgh A 133, 773805.
Crandall, M. G., Ishii, H. and Lions, P.-L. (1992), ‘User's guide to viscosity solutions of second order partial differential equations’, Bull. Amer. Math. Soc. 27, 167.
Maso, G. Dal (1993), An Introduction to Γ -Convergence, Birkhäuser.
Maso, G. Dal and Modica, L. (1986), ‘Nonlinear stochastic homogenization and ergodic theory’, J. Reine Angew. Math. 368, 2842.
Daubechies, I. (1991), Ten Lectures on Wavelets, SIAM.
De Giorgi, E. and Franzoni, T. (1975), ‘Su un tipo di convergenza variationale’, Atti. Acad. Naz. Lincei Rend. Cl. Sci. Mat. 58, 842850.
De Giorgi, E. and Spagnolo, S. (1973), ‘Sulla convergenza degli integrali dell'energia par operatori ellittici del secundo ordine’, Boll. Un. Mat. Ital. 4, 391411.
Dirr, N., Karali, G. and Yip, A. (2007), Pulsating wave for mean curvature flow in homogeneous medium. Preprint.
Dorobantu, M. and Engquist, B. (1998), ‘Wavelet-based numerical homogenization’, SIAM J. Numer. Anal. 35, 540559.
Durlofsky, L. J. (1991), ‘Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media’, Water. Resour. Res. 27, 699708.
Durlofsky, L. J. (1998), ‘Coarse scale models of two-phase flow in heterogeneous reservoirs: Volume averaged equations and their relationship to existing upscaling techniques’, Comput. Geosci. 2, 7392.
W. E, and Engquist, B. (2003 a), ‘The heterogeneous multi-scale method’, Commun. Math. Sci. 1, 87133.
W. E, and Engquist, B. (2003 b), ‘Multi-scale modeling and computation’, Notices Amer. Math. Soc. 50, 10621070.
W. E, , Engquist, B., Li, X., Ren, W. and Vanden-Eijnden, E. (2007), ‘Heterogeneous multiscale methods’, Commun. Comput. Phys. 2, 367450.
Engquist, B. and Luo, E. (1997), ‘Convergence of a multigrid method for elliptic equations with highly oscillatory coefficients’, SIAM J. Numer. Anal. 34, 22542273.
Engquist, B. and Runborg, O. (2001), Wavelet-based numerical homogenization with applications. In Proc. Conference on Multiscale and Multiresolution Methods: Theory and Applications, Vol. 20 of Lecture Notes in Computational Science and Engineering, Springer, pp. 97148.
Engquist, B. and Runborg, O. (2002), Projection generated homogenization. In Proc. Conference on Multiscale Problems in Science and Technology, Springer, pp. 129150.
Evans, L. C. (1989), ‘The perturbed test function method for viscosity solutions of nonlinear PDE’, Proc. Roy. Soc. Edinburgh A 111, 359375.
Evans, L. C. (1992), ‘Periodic homogenization of certain fully nonlinear partial differential equations’, Proc. Roy. Soc. Edinburgh A 120, 245265.
Gilbert, A. C. (1998), ‘A comparison of multiresolution and classical one-dimensional homogenization schemes’, Appl. Comput. Harmon. Anal. 5, 135.
Hou, T. Y. (2003), Numerical approximations to multiscale solutions in partial differential equations. In Frontiers in Numerical Analysis (Blowey, J. F., Craig, A. W. and Shardlow, T., eds), Springer, pp. 241302.
Hou, T. Y. and Wu, X.-H. (1997), ‘A multiscale finite element method for elliptic problems in composite materials and porous media’, J. Comput. Phys. 134, 169189.
Hou, T. Y., Wu, X.-H. and Cai, Z. (1999), ‘Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients’, Math. Comp. 68, 913943.
Hughes, T. J. R., Feijo'o, G. R., Mazzei, L. and Quicy, J.-B. (1999), ‘The variational multiscale method: A paradigm for computational mechanics’, Comput. Methods Appl. Mech. Engrg 166, 515533.
Ishii, H. (1999), Homogenization of the Cauchy problem for Hamilton–Jacobi equations. In Stochastic Analysis, Control, Optimization and Applications, Systems & Control: Foundations & Applications, Birkhäuser, Boston, pp. 305324.
Jikov, V. V., Kozlov, S. M. and Oleinik, O. A. (1991), Homogenization of Differential Operators and Integral Functions, Springer.
Keller, J. B. (1977), Effective behavior of heterogeneous media. In Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, pp. 631644.
Kevrekidis, I. G., Gear, C. W., Hyman, J. M., Kevrekidis, P. G., Runborg, O. and Theodoropoulos, C. (2003), ‘Equation-free, coarse-ground multiscale computation: Enabling microscopic simulators to perform system-level analysis’, Commun. Math. Sci. 1, 715762.
Knapek, S. (1999), ‘Matrix-dependent multigrid-homogenization for diffusion problems’, SIAM J. Sci. Statist. Comput. 20, 512533.
Kosygina, E., Rezakhanlou, F. and Varadhan, S. R. S. (2006), ‘Stochastic homogenization for Hamilton–Jacobi–Bellman equations’, Comm. Pure Appl. Math. 59, 14891521.
Kozlov, S. M. (1985), ‘The method of averaging and walk in inhomogeneous environments’, Russian Math. Surveys 40, 73145.
LeVeque, R. (1990), Numerical Methods for Conservation Laws, Birkhäuser.
Lions, P.-L. and Souganidis, P. E. (2003), ‘Correctors for the homogenization of Hamilton–Jacobi equations in a stationary ergodic setting’, Comm. Pure Appl. Math. LVI, 15011524.
Lions, P.-L. and Souganidis, P. E. (2005 a), ‘Homogenization of degenerate secondorder PDE in periodic and almost periodic environments and applications’, Ann. Inst. H. Poincaré, Anal. Nonlineaire 22, 667677.
Lions, P.-L. and Souganidis, P. E. (2005 b), ‘Homogenization for “viscous” Hamilton–Jacobi equations in stationary, ergodic media’, Comm. Partial Differential Equations 30, 335376.
Lions, P.-L. and Souganidis, P. E. (2008), Homogenization of Hamilton–Jacobi and viscous Hamilton–Jacobi equations in stationary, ergodic environments revisited. Preprint.
Lions, P.-L., Papanicolaou, G. and Varadhan, S. R. S. (1983), Homogenization of Hamilton–Jacobi equations. Unpublished.
Marchenko, V. A. and Khruslov, E. Y. (2006), Homogenization of Partial Differential Equations, Vol. 46 of Progress in Mathematical Physics, Birkhäuser.
Murat, F. and Tartar, L. (1977), Calculus of variations and homogenization. In Topics in the Mathematical Modelling of Composite Materials (Cherkaev, A. and Kohn, R. V., eds), Birkhäuser, Basel, pp. 139173. Originally in French from 1985.
Neuss, N., Jäger, W. and Wittum, G. (2000), ‘Homogenization and multigrid’, Computing 66, 121.
Nguetseng, G. (1989), ‘A general convergence result for a functional related to the theory of homogenization’, SIAM J. Math. Anal. 20, 608623.
Obinata, G. and Anderson, D. O. (2001), Model Reduction for Control System Design, Springer.
Papanicolaou, G. and Varadhan, S. R. S. (1979), Boundary value problems with rapidly oscillating random coefficients. In Proc. Colloq. on Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory (Fritz, J., Lebaritz, J. L. and Szasz, D., eds), Vol. 10 of Colloquia Mathematica Societ. Janos Bolyai, pp. 835873.
Papanicolaou, G. and Varadhan, S. R. S. (1981), Diffusion with random coefficients. In Essays in Statistics and Probability (Krishnaiah, P. R., ed.), North-Holland.
Pavliotis, G. A. and Stewart, A. M. (2007), Multiscale Methods: Averaging and Homogenization, Springer.
Rezankhanlou, F. and Tarver, J. (2000), ‘Homogenization for stochastic Hamilton–Jacobi equations’, Arch. Ration. Mech. Anal. 151, 277309.
Shannon, C. E. (1949), ‘Communication in the presence of noise’, Proc. Inst. Radio Engineers 37, 1021.
Souganidis, P. E. (1999), ‘Stochastic homogenization of Hamilton–Jacobi equations and some applications’, Asympt. Anal. 20, 111.
Tartar, L. (1977), Cours Peccot au Collège de France. Unpublished.
Tartar, L. (1989), Nonlocal effects induced by homogenization. In PDE and Calculus of Variation, Birkhäuser, pp. 925938.
Xu, K. and Prendergast, K. H. (1994), ‘Numerical Navier–Stokes solutions from gas kinetic theory’, J. Comput. Phys. 114, 917.
Yue, X. Y. and W. E, (2008), ‘The local microscale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size’, J. Comput. Phys., to appear.
Yurinskii, V. V. (1980), ‘On the homogenization of boundary value problems with random coeffcients’, Sibir. Matem. Zh. 21, 209223. English translation: Siber. Math. J. 21 (1981), 470–482.
Yurinskii, V. V. (1982), ‘On the homogenization of non-divergent second order equations with random coeffcients’, Sibir. Matem. Zh. 23, 176188. English translation: Siber. Math. J. 23 (1982), 276–287.
Zhikov, V. V. (1993), ‘Asymptotic problems related to a second-order parabolic equation in nondivergence form with randomly homogeneous coeffcients’ (Russian), Differentsial'nye Uravneniya 29, 859869. English translation: Differential Equations 29 (1993), 735–744.
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Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
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