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.
Mots-clés : model reduction, effective equation, multiscale PDE, harmonic coordinates, decomposition
@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 },
pages = {449--474},
publisher = {EDP-Sciences},
volume = {48},
number = {2},
year = {2014},
doi = {10.1051/m2an/2013115},
mrnumber = {3177853},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013115/}
}
TY - JOUR AU - Ci, Maolin AU - Hou, Thomas Y. AU - Shi, Zuoqiang TI - A Multiscale Model Reduction Method for Partial Differential Equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 449 EP - 474 VL - 48 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013115/ DO - 10.1051/m2an/2013115 LA - en ID - M2AN_2014__48_2_449_0 ER -
%0 Journal Article %A Ci, Maolin %A Hou, Thomas Y. %A Shi, Zuoqiang %T A Multiscale Model Reduction Method for Partial Differential Equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 449-474 %V 48 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013115/ %R 10.1051/m2an/2013115 %G en %F 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 , Multiscale problems and techniques. Special Issue, Tome 48 (2014) no. 2, pp. 449-474. doi : 10.1051/m2an/2013115. http://archive.numdam.org/articles/10.1051/m2an/2013115/
[1] and , Univalent σ-harmonic mappings. Arch. Rational Mech. Anal. 158 (2001) 155-171. | MR | Zbl
[2] and , A multiscale finite element method for numerical homogenization. SIAM MMS 4 (2005) 790-812. | MR | Zbl
[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 | Zbl
[4] , , and , A multiscale mortar mixed finite element method. SIAM MMS 6 (2007) 319-346. | MR | Zbl
[5] and , Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods. SIAM J. Numer. Anal. 20 (1983) 510-536. | MR | Zbl
[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 | Zbl
[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 | Zbl
[8] , and , Asymptotic Analysis for Periodic Structure. North-Holland, Amsterdam (1978). | MR | Zbl
[9] , , , , and , Reduced Basis Techniques for Stochastic Problems. Arch. Comput. Meth. Eng. 17 (2012) 435-454. | MR | Zbl
[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 | Zbl
[12] , and , A New multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79 (2010) 1915-1955. | MR | Zbl
[13] and , The heterogeneous multi-scale methods. Commun. Math. Sci. 1 (2003) 87-133. | MR | Zbl
[14] , and , Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937-955. | MR
[15] , , and , Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220 (2006) 155-174. | MR | Zbl
[16] and , Multiscale finite element methods. Theory and applications. Springer (2009). | MR | Zbl
[17] , and , Convergence of a nonconforming multiscale finite element method. SIAM J. Num. Anal. 37 (2000) 888-910. | MR | Zbl
[18] , and , Generalized multiscale finite element methods (GMsFEM). Accepted by JCP (2013). | MR
[19] and , Domain decomposition preconditioners for multiscale flows in high-contrast media: Reduced dimension coarse spaces. SIAM MMS 8 (2009) 1621-1644. | MR
[20] , and , Domain decomposition for multiscale PDEs. Numer. Math. 106 (2007) 589-626. | MR | Zbl
[21] and , A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR | Zbl
[22] , and , Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913-943. | MR | Zbl
[23] , , and , The variational multiscale method - a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24. | MR | Zbl
[24] , and , Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47-67. | Zbl
[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 | Zbl
[27] , and , Convetion of mircrostructure and related problems. SIAM J. Appl. Math. 45 (1985) 780-797. | MR | Zbl
[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 | Zbl
[29] and , Metric based up-scaling. Commun. Pure Appl. Math. LX (2007) 675-723. | MR | Zbl
[30] and , Homogenization of parabolic equations with a continuum of space and time scales. SIAM J. Numer. Anal. 46 (2007) 1-36. | MR | Zbl
[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
Cité par Sources :
