Journal of the American Mathematical Society

Analysis of the heterogeneous multiscale method for elliptic homogenization problems

Author(s): Weinan E; Pingbing Ming; Pingwen Zhang
Journal: J. Amer. Math. Soc. 18 (2005), 121-156.
MSC (2000): Primary 65N30, 74Q05; Secondary 74Q15, 65C30.
Posted: September 16, 2004
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Abstract: A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

1.: R.A. Adams and J.J. F. Fournier, Sobolev Spaces, second edition, Academic Press, New York, 2003.
2.: G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23(1992), 1482-1518. MR 1185639 (93k:35022)
3.: I. Babuska, Homogenization and its applications, mathematical and computational problems, Numerical Solutions of Partial Differential Equations-III (SYNSPADE 1975, College Park, MD, May 1975) (B. Hubbard ed.), Academic Press, New York, 1976, pp. 89-116.MR 0502025 (58:19215)
4.: I. Babuska, Solution of interface by homogenization. I, II, III, SIAM J. Math. Anal., 7 (1976), 603-634, 635-645, 8 (1977), 923-937.MR 0509273 (58:23013a); MR 0509277 (58:23013b); MR 0509282 (58:23014)
5.: A. Bensoussan, J.L. Lions and G.C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.MR 0503330 (82h:35001)
6.: L. Boccardo and F. Murat, Homogénéisation de problémes quasi-linéaires, Publ. IRMA, Lille., 3 (1981), no. 7, 13-51.
7.: J.F. Bourgat, Numerical experiments of the homogenization method for operators with periodic coefficients, Lecture Notes in Mathematics, Vol. 707. Springer-Verlag, 1977, pp. 330-356. MR 0540121 (80e:65108)
8.: A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp., 31 (1977), 333-390. MR 0431719 (55:4714)
9.: S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994. MR 1278258 (95f:65001)
10.: P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
11.: P.G. Ciarlet and P.-A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, The Mathematics Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz ed.), Academic Press, Inc., 1972, pp. 409-474.MR 0421108 (54:9113)
12.: P. Clément, Approximation by finite element functions using local regularization, RAIRO., 9 (1975), 77-84. MR 0400739 (53:4569)
13.: J.G. Conlon and A. Naddaf, On homogenization of elliptic equations with random coefficients, Electron J. Probab., 5 (2000), 1-58. MR 1768843 (2002j:35328)
14.: L.J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous poros-media, Water. Resour. Res., 28 (1992), 699-708.
15.: W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math., 45 (1992), 301-326. MR 1151269 (92k:35026)
16.: W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1(2003), 87-132. MR 1979846 (2004b:35019)
17.: W. E and B. Engquist, The heterogeneous multiscale method for homogenization problems, submitted to MMS, 2002.
18.: W. E and B. Engquist, Multiscale modeling and computation, Notice Amer. Math. Soc., 50 (2003), 1062-1070. MR 2002752
19.: W. E and X.Y. Yue, Heterogeneous multiscale method for locally self-similar problems, Comm. Math. Sci., 2 (2004), 137-144.
20.: Y.R. Efendiev, T. Hou and X. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), 888-910. MR 1740386 (2002a:65176)
21.: M.I. Freidlin, Functional Integration and Partial Differential Equations, Princeton, NJ, 1985. MR 0833742 (87g:60066)
22.: M.I. Freidlin and A.D. Wentzell, Random Perturbations of Dynamical Systems, second edition, Springer-Verlag, 1998. MR 1652127 (99h:60128)
23.: N. Fusco and G. Moscarielio, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura. Appl., 146 (1987), 1-13. MR 0916685 (89a:35025)
24.: D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, second edition, Springer-Verlag, New York, 1983. MR 0737190 (86c:35035)
25.: T. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), 169-189. MR 1455261 (98e:73132)
26.: I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Comm. Math. Sci., 1 (2003), 715-762.MR 2041455
27.: J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. Solids., 49 (2001), 1899-1923.
28.: S.M. Kozlov, Homogenization of random operators, Math. USSR Sb., 37 (1980), 167-180. MR 0542557 (81m:35142)
29.: A.M. Matache, I. Babuska and C. Schwab, Generalized -FEM in homogenization, Numer. Math., 86 (2000), 319-375.MR 1777492 (2001f:65139)
30.: G.W. Milton, The Theory of Composites, Cambridge University Press, 2002.MR 1899805 (2003d:74077)
31.: P.B. Ming and X.Y. Yue, Numerical methods for multiscale elliptic problems, preprint, 2003.
32.: S. Moskow and M. Vogelius, First order corrections to the homogenized eigenvalues of a periodic composite medium, Proc. Royal Soc. Edinburgh. Sec. A., 127 (1997), 1263-1299. MR 1489436 (99g:35018)
33.: F. Murat and L. Tartar, H-Convergence, Cours Peccot, 1977. Reprinted in Topics in the Mathematical Modeling of Composite Materials (A. Cherkaev and R. Kohn eds.), Birkhäuser, Boston, 1997, pp. 21-43.MR 1493039
34.: G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.MR 0990867 (90j:35030)
35.: J.T. Oden and K.S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms, J. Comput. Phys., 164 (2000), 22-47.MR 1786241 (2001e:74079)
36.: G.C. Papanicolaou and S.R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, at Proceedings of the conference on Random Fields, Esztergom, Hungary, 1979, published in Seria Colloquia Mathematica Societatis Janos Bolyai, North-Holland, 27 (1981), pp. 835-873. MR 0712714 (84k:58233)
37.: R. Rannacher and R. Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp., 38 (1982), 437-445. MR 0645661 (83e:65180)
38.: C. Schwab and A.-M. Matache, Generalzied FEM for homogenization problems, Multiscale and Multiresolution Methods: Theory and Applications, Lecture Notes in Computational Sciences and Engineering, Vol. 20 (T. J. Barth, T. Chan and R. Heimes eds.), Springer-Verlag, Heidelberg, 2002, pp. 197-237. MR 1928567 (2003i:65115)
39.: R. Scott, Optimal L $^\infty$ estimates for the finite element method on irregular meshes, Math. Comp., 30 (1976), 681-697.MR 0436617 (55:9560)
40.: S. Spagnolo, Convergence in energy for elliptic operators, Numerical Solutions of Partial Differential Equations-III (SYNSPADE 1975, College Park, MD, May 1975) (B. Hubbard ed.), Academic Press, New York, 1976, pp. 469-499. MR 0477444 (57:16971)
41.: L. Tartar, An introduction to the homogenization method in optimal design, Optimal shape Design (A. Cellina and A. Ornelas eds.), Lecture Notes in Mathematics, Vol. 1740. Springer-verlag, 2000. pp. 47-156.MR 1804685
42.: J. Xu, Two-grid discretization techniques for linear and nonlinear PDE, SIAM J. Numer. Anal., 33 (1996), 1759-1777.MR 1411848 (97i:65169)
43.: V.V. Yurinskii, Averaging of symmetric diffusion in random media, Sibirsk. Mat. Zh. 23 (1982), no. 2, 176-188; English transl. in Siberian Math J., 27 (1986), 603-613. MR 0867870 (88e:35190)
44.: V.V. Zhikov, On an extension of the method of two-scale convergence and its application, Sbornik: Mathematics, 191 (2000), 973-1014. MR 1809928 (2001k:35026)
45.: V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Heidelberg, 1994.MR 1329546 (96h:35003b)

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 65N30, 74Q05, 74Q15, 65C30.

Weinan E
Affiliation: Department of Mathematics and PACM, Princeton University, Princeton, New Jersey 08544 and School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
Email: weinan@Princeton.EDU

Pingbing Ming
Affiliation: No. 55, Zhong-Guan-Cun East Road, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, People's Republic of China
Email: mpb@lsec.cc.ac.cn

Pingwen Zhang
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
Email: pzhang@pku.edu.cn

DOI: 10.1090/S0894-0347-04-00469-2
PII: S 0894-0347(04)00469-2
Keywords: Heterogeneous multiscale method, homogenization problems.
Received by editor(s): January 2, 2003
Posted: September 16, 2004
Additional Notes: The work of the first author was partially supported by ONR grant N00014-01-1-0674 and the National Natural Science Foundation of China through a Class B Award for Distinguished Young Scholars 10128102.
The work of the second author was partially supported by the Special Funds for the Major State Basic Research Projects G1999032804 and was also supported by the National Natural Science Foundation of China 10201033.
The work of the third author was partially supported by the Special Funds for the Major State Research Projects G1999032804 and the National Natural Science Foundation of China for Distinguished Young Scholars 10225103.
We thank Bjorn Engquist for inspiring discussions on the topic studied here.
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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