Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast

Abstract

We construct finite-dimensional approximations of solution spaces of divergence-form operators with L-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in H1 if source terms are in the unit ball of L2 instead of the unit ball of H-1. Approximation spaces are generated by solving elliptic PDEs on localized subdomains with source terms corresponding to approximation bases for H2. The H1-error estimates show that 𝒪(h-d)-dimensional spaces with basis elements localized to subdomains of diameter 𝒪(hαln1h) (with α[12,1)) result in an 𝒪(h2-2α) accuracy for elliptic, parabolic, and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved, provided that localized subdomains contain buffer zones of width 𝒪(hαln1h), where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

MSC codes

  1. 34E13
  2. 35B27

Keywords

  1. homogenization
  2. localization
  3. cell problem
  4. high contrast
  5. continuum of scales

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References

1.
A. Abdulle and M. J. Grote, Finite element heterogeneous multiscale method for the wave equation, Multiscale Model. Simul., 9 (2011), pp. 766–792.
2.
G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Model. Simul., 4 (2005), pp. 790–812.
3.
T. Arbogast and K. J. Boyd, Subgrid upscaling and mixed multiscale finite elements, SIAM J. Numer. Anal., 44 (2006), pp. 1150–1171.
4.
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), pp. 1279–1305.
5.
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), pp. 945–981.
6.
I. Babuška and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul., 9 (2011), pp. 373–406.
7.
I. Babuška and J. E. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods, SIAM J. Numer. Anal., 20 (1983), pp. 510–536.
8.
G. Bal and W. Jing, Corrector theory for msfem and hmm in random media, Multiscale Model. Simul., to appear.
9.
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North–Holland, Amsterdam, 1978.
10.
L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Arch. Ration. Mech. Anal., 198 (2010), pp. 677–721.
11.
C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients, Numer. Math., 85 (2000), pp. 579–608.
12.
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), pp. 717–724.
13.
X. Blanc, C. Le Bris and P.-L. Lions, Stochastic homogenization and random lattices, J. Math. Pures Appl. (9), 88 (2007), pp. 34–63.
14.
D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, London, 2007.
15.
A. Braides, Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University Press, Oxford, 2002.
16.
L. V. Branets, S. S. Ghai, S. L. Lyons and X.-H. Wu, Challenges and technologies in reservoir modeling, Commun. Comput. Phys., 6 (2009), pp. 1–23.
17.
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Texts Appl. Math. 15, Springer, New York, 2002.
18.
D. L. Brown, A note on the numerical solution of the wave equation with piecewise smooth coefficients, Math. Comp., 42 (1984), pp. 369–391.
19.
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), pp. 1–17.
20.
E. Cancès, C. Le Bris, Y. Maday, N. C. Nguyen, A. T. Patera and G. S. H. Pau, Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry, in High-Dimensional Partial Differential Equations in Science and Engineering, CRM Proc. Lecture Notes 41, AMS, Providence, RI, 2007, pp. 15–47.
21.
K. D. Cherednichenko, V. P. Smyshlyaev and V. V. Zhikov, Non-local homogenized limits for composite media with highly anisotropic periodic fibres, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), pp. 87–114.
22.
C.-C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), pp. 1915–1955.
23.
A. Doostan and H. Owhadi, A non-adapted sparse approximation of pdes with stochastic inputs, J. Comput. Phys., 230 (2011), pp. 3015–3034.
24.
W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2 (2007), pp. 367–450.
25.
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), pp. 937–955.
26.
Y. Efendiev, V. Ginting, T. Hou and R. Ewing, Accurate multiscale finite element methods for two-phase flow simulations, J. Comput. Phys., 220 (2006), pp. 155–174.
27.
Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications, Appl. Numer. Math., 57 (2007), pp. 577–596.
28.
B. Engquist, H. Holst and O. Runborg, Multi-scale methods for wave propagation in heterogeneous media, Commun. Math. Sci., 9 (2011), pp. 33–56.
29.
B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numer., 17 (2008), pp. 147–190.
30.
B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation, Comm. Pure Appl. Math., 64 (2011), pp. 697–735.
31.
B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: Moving perfectly matched layers, Multiscale Model. Simul., 9 (2011), pp. 686–710.
32.
E. D. Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell’aera, Rend. Mat. Appl. (7), 8 (1975), pp. 277–294.
33.
E. D. Giorgi, New problems in Γ-convergence and G-convergence, in Free Boundary Problems, Vol. II, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980, pp. 183–194.
34.
A. Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies, Multiscale Model. Simul., 5 (2006), pp. 996–1043.
35.
A. Gloria, Reduction of the resonance error. Part 1: Approximation of homogenized coefficients, Math. Models Methods Appl. Sci., 21 (2011), pp. 1601–1630.
36.
A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., to appear.
37.
H. Harbrecht, R. Schneider and C. Schwab, Sparse second moment analysis for elliptic problems in stochastic domains, Numer. Math., 109 (2008), pp. 385–414.
38.
K. Höllig, Finite Element Methods with B-Splines, Frontiers Appl. Math. 26, SIAM, Philadelphia, PA, 2003.
39.
K. Höllig, U. Reif and J. Wipper, Weighted extended B-spline approximation of Dirichlet problems, SIAM J. Numer. Anal., 39 (2001), pp. 442–462.
40.
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), pp. 169–189.
41.
T. Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), pp. 913–943.
42.
S. M. Kozlov, The averaging of random operators, Mat. Sb., 109 (1979), pp. 188–202.
43.
L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera and D. V. Rovas, Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Math. Acad. Sci. Paris, 331 (2000), pp. 153–158.
44.
J. M. Melenk, On n-widths for elliptic problems, J. Math. Anal. Appl., 247 (2000), pp. 272–289.
45.
P. Ming and X. Yue, Numerical methods for multiscale elliptic problems, J. Comput. Phys., 214 (2006), pp. 421–445.
46.
F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), pp. 489–507.
47.
F. Murat and L. Tartar, H-convergence, Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger, 1978.
48.
J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems, Multiscale Model. Simul., 7 (2008), pp. 171–196.
49.
H. Owhadi and L. Zhang, Metric-based upscaling, Comm. Pure Appl. Math., 60 (2007), pp. 675–723.
50.
H. Owhadi and L. Zhang, Homogenization of parabolic equations with a continuum of space and time scales, SIAM J. Numer. Anal., 46 (2007), pp. 1–36.
51.
H. Owhadi and L. Zhang, Homogenization of the acoustic wave equation with a continuum of scales, Comput. Methods Appl. Mech. Engrg., 198 (2008), pp. 97–406.
52.
H. Owhadi and L. Zhang, Numerical Homogenization with Localized Bases, http://www.youtube.com/view_play_list?p=2009D30DF7B07294 (2010).
53.
G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in Random Fields, Vol. I, II, Colloq. Math. Soc. János Bolyai 27, North–Holland, Amsterdam, 1981, pp. 835–873.
54.
A. Pinkus, n-Width in Approximation Theory, Springer, New York, 1985.
55.
A. Pinkus, n-Widths in Approximation Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 7, Springer, Berlin, 1985.
56.
A. Sei and W. W. Symes, Error analysis of numerical schemes for the wave equation in heterogeneous media, Appl. Numer. Math., 15 (1994), pp. 465–480.
57.
S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche., Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22 (1968), pp. 571–597.
58.
S. Spagnolo, Convergence in energy for elliptic operators, in Numerical Solution of Partial Differential Equations, III, Academic Press, New York, 1976, pp. 469–498.
59.
W. W. Symes and T. Vdovina, Interface error analysis for numerical wave propagation, Comput. Geosci., 13 (2009), pp. 363–371.
60.
R. A. Todor and C. Schwab, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal., 27 (2007), pp. 232–261.
61.
C. D. White and R. N. Horne, Computing absolute transmissibility in the presence of finescale heterogeneity, in Proceedings of the SPE Symposium on Reservoir Simulation, 1987, p. 16011.
62.
X. H. Wu, Y. Efendiev and T. Y. Hou, Analysis of upscaling absolute permeability, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), pp. 185–204.
63.
V. V. Yurinskii, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh., 27 (1986), pp. 167–180.
64.
L. Zhang, L. Berlyand, M. Federov and H. Owhadi, Global energy matching method for atomistic-to-continuum modeling of self-assembling biopolymer aggregates, Multiscale Model. Simul., 8 (2010), pp. 1958–1980.

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Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1373 - 1398
ISSN (online): 1540-3467

History

Submitted: 8 November 2010
Accepted: 3 August 2011
Published online: 1 November 2011

MSC codes

  1. 34E13
  2. 35B27

Keywords

  1. homogenization
  2. localization
  3. cell problem
  4. high contrast
  5. continuum of scales

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