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Analysis of the heterogeneous multiscale method for elliptic homogenization problems. (English) Zbl 1060.65118

The heterogeneous multiscale method (HMM) is a general methodology for designing sublinear algorithms by expliting scale separation and other special features of the problem. It consists of two components: selection of a macroscopic solver and estimating the missing macroscale data by solving locally the fine scale problem. This paper estimates the error between the numerical solutions of HMM and the solutions of the problem \(-\text{div} (A(x)\nabla U(x))=f(x)\), \(x\in D\), \(U(x)=0\), \(x\in\partial D\). The authors prove a general statement that this error is controlled by a standard error in the macroscale solver plus a new term due to the error in estimating the stiffness matrix. This second part is only done for either periodic or random homogenisation problems.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65C30 Numerical solutions to stochastic differential and integral equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J25 Boundary value problems for second-order elliptic equations
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[1] R.A. Adams and J.J. F. Fournier, Sobolev Spaces, second edition, Academic Press, New York, 2003. · Zbl 1098.46001
[2] Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), no. 6, 1482 – 1518. · Zbl 0770.35005
[3] Ivo Babuška, Homogenization and its application. Mathematical and computational problems, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 89 – 116.
[4] Ivo Babuška, Solution of interface problems by homogenization. I, SIAM J. Math. Anal. 7 (1976), no. 5, 603 – 634. , https://doi.org/10.1137/0507048 Ivo Babuška, Solution of interface problems by homogenization. II, SIAM J. Math. Anal. 7 (1976), no. 5, 635 – 645. , https://doi.org/10.1137/0507049 Ivo Babuška, Solution of interface problems by homogenization. I, SIAM J. Math. Anal. 7 (1976), no. 5, 603 – 634. , https://doi.org/10.1137/0507048 Ivo Babuška, Solution of interface problems by homogenization. II, SIAM J. Math. Anal. 7 (1976), no. 5, 635 – 645. , https://doi.org/10.1137/0507049 Ivo Babuška, Solution of interface problems by homogenization. III, SIAM J. Math. Anal. 8 (1977), no. 6, 923 – 937. · Zbl 0402.35046
[5] Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. · Zbl 0404.35001
[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, Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977) Lecture Notes in Math., vol. 704, Springer, Berlin, 1979, pp. 330 – 356. · Zbl 0405.65062
[8] Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333 – 390. · Zbl 0373.65054
[9] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. · Zbl 0804.65101
[10] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[11] P. G. Ciarlet and P.-A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 409 – 474. · Zbl 0262.65070
[12] Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique 9 (1975), no. R-2, 77 – 84 (English, with Loose French summary). · Zbl 0368.65008
[13] Joseph G. Conlon and Ali Naddaf, On homogenization of elliptic equations with random coefficients, Electron. J. Probab. 5 (2000), no. 9, 58. · Zbl 0956.35013
[14] L.J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous poros-media, Water. Resour. Res., 28 (1992), 699-708.
[15] Weinan E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45 (1992), no. 3, 301 – 326. · Zbl 0794.35014
[16] Weinan E and Bjorn Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87 – 132. · Zbl 1093.35012
[17] W. E and B. Engquist, The heterogeneous multiscale method for homogenization problems, submitted to MMS, 2002.
[18] Weinan E and Bjorn Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc. 50 (2003), no. 9, 1062 – 1070. · Zbl 1032.65013
[19] W. E and X.Y. Yue, Heterogeneous multiscale method for locally self-similar problems, Comm. Math. Sci., 2 (2004), 137-144. · Zbl 1081.82563
[20] Yalchin R. Efendiev, Thomas Y. Hou, and Xiao-Hui Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37 (2000), no. 3, 888 – 910. · Zbl 0951.65105
[21] Mark Freidlin, Functional integration and partial differential equations, Annals of Mathematics Studies, vol. 109, Princeton University Press, Princeton, NJ, 1985. · Zbl 0568.60057
[22] M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1998. Translated from the 1979 Russian original by Joseph Szücs. · Zbl 0922.60006
[23] N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl. (4) 146 (1987), 1 – 13. · Zbl 0636.35027
[24] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[25] Thomas Y. Hou and Xiao-Hui Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), no. 1, 169 – 189. · Zbl 0880.73065
[26] Ioannis G. Kevrekidis, C. William Gear, James M. Hyman, Panagiotis G. Kevrekidis, Olof Runborg, and Constantinos Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci. 1 (2003), no. 4, 715 – 762. · Zbl 1086.65066
[27] J. Knap and M. Ortiz, An analysis of the quasicontinuum method, J. Mech. Phys. Solids., 49 (2001), 1899-1923. · Zbl 1002.74008
[28] S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.) 109(151) (1979), no. 2, 188 – 202, 327 (Russian).
[29] A. M. Matache, I. Babuška, and C. Schwab, Generalized \?-FEM in homogenization, Numer. Math. 86 (2000), no. 2, 319 – 375. · Zbl 0964.65125
[30] Graeme W. Milton, The theory of composites, Cambridge Monographs on Applied and Computational Mathematics, vol. 6, Cambridge University Press, Cambridge, 2002. · Zbl 0993.74002
[31] P.B. Ming and X.Y. Yue, Numerical methods for multiscale elliptic problems, preprint, 2003.
[32] Shari Moskow and Michael Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1263 – 1299. · Zbl 0888.35011
[33] François Murat and Luc Tartar, \?-convergence, Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., vol. 31, Birkhäuser Boston, Boston, MA, 1997, pp. 21 – 43. · Zbl 0920.35019
[34] Gabriel Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), no. 3, 608 – 623. · Zbl 0688.35007
[35] J. Tinsley Oden and Kumar 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), no. 1, 22 – 47. · Zbl 0992.74072
[36] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random fields, Vol. I, II (Esztergom, 1979) Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam-New York, 1981, pp. 835 – 873. · Zbl 0499.60059
[37] Rolf Rannacher and Ridgway Scott, Some optimal error estimates for piecewise linear finite element approximations, Math. Comp. 38 (1982), no. 158, 437 – 445. · Zbl 0483.65007
[38] Christoph Schwab and Ana-Maria Matache, Generalized FEM for homogenization problems, Multiscale and multiresolution methods, Lect. Notes Comput. Sci. Eng., vol. 20, Springer, Berlin, 2002, pp. 197 – 237. · Zbl 0989.65131
[39] Ridgway Scott, Optimal \?^{\infty } estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681 – 697. · Zbl 0349.65060
[40] Sergio Spagnolo, Convergence in energy for elliptic operators, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 469 – 498.
[41] Luc Tartar, An introduction to the homogenization method in optimal design, Optimal shape design (Tróia, 1998) Lecture Notes in Math., vol. 1740, Springer, Berlin, 2000, pp. 47 – 156. · Zbl 1040.49022
[42] Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759 – 1777. · Zbl 0860.65119
[43] V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh. 27 (1986), no. 4, 167 – 180, 215 (Russian).
[44] V. V. Zhikov, On an extension and an application of the two-scale convergence method, Mat. Sb. 191 (2000), no. 7, 31 – 72 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 7-8, 973 – 1014. · Zbl 0969.35048
[45] Усреднение дифференциал\(^{\приме}\)ных операторов, ”Наука”, Мосцощ, 1993 (Руссиан, щитх Енглиш анд Руссиан суммариес). В. В. Јиков, С. М. Козлов, анд О. А. Олейник, Хомогенизатион оф дифферентиал операторс анд интеграл фунцтионалс, Спрингер-Верлаг, Берлин, 1994. Транслатед фром тхе Руссиан бы Г. А. Ыосифиан [Г. А. Иосиф\(^{\приме}\)ян].
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