We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show that the solution to the effective equation is in H2, and can be approximated by a regular coarse mesh. When the multiscale problem has scale separation and a periodic structure, our method recovers the traditional homogenized equation. Furthermore, we provide error analysis for our method and show that the solution to the effective equation is close to the original multiscale solution in the H1 norm. Numerical results are presented to demonstrate the accuracy and robustness of the proposed method for several multiscale problems without scale separation, including a problem with a high contrast coefficient.
@article{M2AN_2014__48_2_449_0,
author = {Ci, Maolin and Hou, Thomas Y. and Shi, Zuoqiang},
title = {A Multiscale Model Reduction Method for Partial Differential Equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {449-474},
doi = {10.1051/m2an/2013115},
mrnumber = {3177853},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2014__48_2_449_0}
}
Ci, Maolin; Hou, Thomas Y.; Shi, Zuoqiang. A Multiscale Model Reduction Method for Partial Differential Equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 449-474. doi : 10.1051/m2an/2013115. http://gdmltest.u-ga.fr/item/M2AN_2014__48_2_449_0/
[1] and , Univalent σ-harmonic mappings. Arch. Rational Mech. Anal. 158 (2001) 155-171. | MR 1838656 | Zbl 0977.31006
[2] and , A multiscale finite element method for numerical homogenization. SIAM MMS 4 (2005) 790-812. | MR 2203941 | Zbl 1093.35007
[3] , Some results and examples about the behavior of harmonic functions and Green's funtions with respect to second order elliptic operators. Nagoya Math. J. 165 (2002) 123-158. | MR 1892102 | Zbl 1028.31003
[4] , , and , A multiscale mortar mixed finite element method. SIAM MMS 6 (2007) 319-346. | MR 2306414 | Zbl 1322.76039 | Zbl pre05255539
[5] and , Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods. SIAM J. Numer. Anal. 20 (1983) 510-536. | MR 701094 | Zbl 0528.65046
[6] , and , Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945-981. | MR 1286212 | Zbl 0807.65114
[7] , , and , An Empirical Interpolation Method: Application to Efficient Reduced-Basis Discretization of Partial Differential Equations. C.R. Acad. Sci. Paris Series I 339 (2004) 667-672. | MR 2103208 | Zbl 1061.65118
[8] , and , Asymptotic Analysis for Periodic Structure. North-Holland, Amsterdam (1978). | MR 503330 | Zbl 0404.35001
[9] , , , , and , Reduced Basis Techniques for Stochastic Problems. Arch. Comput. Meth. Eng. 17 (2012) 435-454. | MR 2739947 | Zbl 1269.65005
[10] , , and , A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Advances in Water Resources 26 (2003) 1041-1060.
[11] and , A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72 (2002) 541-576. | MR 1954956 | Zbl 1017.65088
[12] , and , A New multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79 (2010) 1915-1955. | MR 2684351 | Zbl 1202.65154
[13] and , The heterogeneous multi-scale methods. Commun. Math. Sci. 1 (2003) 87-133. | MR 1979846 | Zbl 1093.35012
[14] , and , Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937-955. | MR 2753343 | Zbl pre05867068
[15] , , and , Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220 (2006) 155-174. | MR 2281625 | Zbl 1158.76349
[16] and , Multiscale finite element methods. Theory and applications. Springer (2009). | MR 2477579 | Zbl 1163.65080
[17] , and , Convergence of a nonconforming multiscale finite element method. SIAM J. Num. Anal. 37 (2000) 888-910. | MR 1740386 | Zbl 0951.65105
[18] , and , Generalized multiscale finite element methods (GMsFEM). Accepted by JCP (2013). | MR 3094911
[19] and , Domain decomposition preconditioners for multiscale flows in high-contrast media: Reduced dimension coarse spaces. SIAM MMS 8 (2009) 1621-1644. | MR 2728702 | Zbl pre05869382
[20] , and , Domain decomposition for multiscale PDEs. Numer. Math. 106 (2007) 589-626. | MR 2317926 | Zbl 1141.65084
[21] and , A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR 1455261 | Zbl 0880.73065
[22] , and , Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913-943. | MR 1642758 | Zbl 0922.65071
[23] , , and , The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24. | MR 1660141 | Zbl 1017.65525
[24] , and , Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67. | Zbl 1047.76538
[25] , Proceedings of the International Conference Math., Madrid. European Mathematical Society, Zurich (2006).
[26] , and . Elliptic and parabolic equations with discontinuous coefficients. Math. Research 109, Wiley-VCH (2000). | MR 2260015 | Zbl 0958.35002
[27] , and , Convetion of mircrostructure and related problems. SIAM J. Appl. Math. 45 (1985) 780-797. | MR 804006 | Zbl 0622.76062
[28] and , First-oder corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc. Roy. Soc. Edinburgh. 127A (1997) 1263-1299. | MR 1489436 | Zbl 0888.35011
[29] and , Metric based up-scaling. Commun. Pure Appl. Math. LX (2007) 675-723. | MR 2292954 | Zbl 1190.35070
[30] and , Homogenization of parabolic equations with a continuum of space and time scales. SIAM J. Numer. Anal. 46 (2007) 1-36. | MR 2377253 | Zbl 1170.34037
[31] , and , Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR 2430350 | Zbl pre05344486