Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T06:34:25.280Z Has data issue: false hasContentIssue false
  • English
  • Français

A Multiscale Model Reduction Method for Partial DifferentialEquations

Published online by Cambridge University Press:  20 February 2014

Maolin Ci ,
Thomas Y. Hou  and
Zuoqiang Shi
Maolin Ci
Affiliation:
Applied and Comput. Math, Caltech, Pasadena, CA 91125, USA.. mci@cms.caltech.edu; hou@cms.caltech.edu
Thomas Y. Hou
Affiliation:
Applied and Comput. Math, Caltech, Pasadena, CA 91125, USA.. mci@cms.caltech.edu; hou@cms.caltech.edu
Zuoqiang Shi
Affiliation:
Math Science Center, Tsinghua Univ, Beijing 100084, China.; zqshi@math.tsinghua.edu.cn
Get access

Abstract

We propose a multiscale model reduction method for partial differential equations. Themain purpose of this method is to derive an effective equation for multiscale problemswithout scale separation. An essential ingredient of our method is to decompose theharmonic coordinates into a smooth part and a highly oscillatory part so that the smoothpart is invertible and the highly oscillatory part is small. Such a decomposition plays akey role in our construction of the effective equation. We show that the solution to theeffective equation is in H2, and can be approximated by a regularcoarse mesh. When the multiscale problem has scale separation and a periodic structure,our method recovers the traditional homogenized equation. Furthermore, we provide erroranalysis for our method and show that the solution to the effective equation is close tothe original multiscale solution in the H1 norm. Numerical results are presentedto demonstrate the accuracy and robustness of the proposed method for several multiscaleproblems without scale separation, including a problem with a high contrastcoefficient.

Keywords

Model reductioneffective equationmultiscale PDEharmonic coordinatesdecomposition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 2: Multiscale problems and techniques , March 2014 , pp. 449 - 474
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, G. and Nesi, V., Univalent σ-harmonic mappings. Arch. Rational Mech. Anal. 158 (2001) 155–171. Google Scholar
Allaire, G. and Brizzi, R., A multiscale finite element method for numerical homogenization. SIAM MMS 4 (2005) 790–812. Google Scholar
Ancona, A., 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. Google Scholar
Arbogast, T., Pencheva, G., Wheeler, M.F. and Yotov, I., A multiscale mortar mixed finite element method. SIAM MMS 6 (2007) 319–346. Google Scholar
Babuška, I. and Osborn, E., Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods. SIAM J. Numer. Anal. 20 (1983) 510–536. Google Scholar
Babuška, I., Caloz, G. and Osborn, E., Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945–981. Google Scholar
Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T., 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. Google Scholar
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure. North-Holland, Amsterdam (1978).
Boyaval, S., LeBris, C., Lelièvre, T., Maday, Y., Nguyen, N. and Patera, A., Reduced Basis Techniques for Stochastic Problems. Arch. Comput. Meth. Eng. 17 (2012) 435–454. Google Scholar
Chen, Y., Durlofsky, L.J., Gerritsen, M. and Wen, X.H., A coupled local-global upscaling approach for simulating flow in highly heterogeneous formations. Advances in Water Resources 26 (2003) 1041–1060. Google Scholar
Chen, Z. and Hou, T.Y., A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72 (2002) 541–576. Google Scholar
Chu, C.C., Graham, I. and Hou, T.Y., A New multiscale finite element method for high-contrast elliptic interface problems. Math. Comput. 79 (2010) 1915–1955. Google Scholar
E, Weinan and Engquist, B., The heterogeneous multi-scale methods. Commun. Math. Sci. 1 (2003) 87–133. Google Scholar
Efendiev, Y., Galvis, J. and Wu, X.H., Multiscale finite element methods for high-contrast problems using local spectral basis functions. J. Comput. Phys. 230 (2011) 937–955. Google Scholar
Efendiev, Y., Ginting, V., Hou, T. and Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations. J. Comput. Phys. 220 (2006) 155–174. Google Scholar
Y. Efendiev and T. Hou, Multiscale finite element methods. Theory and applications. Springer (2009).
Efendiev, Y., Hou, T.Y. and Wu, X.H., Convergence of a nonconforming multiscale finite element method. SIAM J. Num. Anal. 37 (2000) 888–910. Google Scholar
Y. Efendiev, J. Galvis and T.Y. Hou, Generalized multiscale finite element methods (GMsFEM). Accepted by JCP (2013).
Galvis, J. and Efendiev, Y., Domain decomposition preconditioners for multiscale flows in high-contrast media: Reduced dimension coarse spaces. SIAM MMS 8 (2009) 1621–1644. Google Scholar
Graham, I.G., Lechner, P.O. and Scheichl, R., Domain decomposition for multiscale PDEs. Numer. Math. 106 (2007) 589–626. Google Scholar
Hou, T.Y. and Wu, X.H., A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. Google Scholar
Hou, T.Y., Wu, X.H. and Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913–943. Google Scholar
Hughes, T., Feijoo, G., Mazzei, L. and Quincy, J., The variational multiscale method – a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24. Google Scholar
Jenny, P., Lee, S.H. and Tchelepi, H., Multi-scale finite volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187 (2003) 47–67. Google Scholar
Y. Maday, Proceedings of the International Conference Math., Madrid. European Mathematical Society, Zurich (2006).
A. Maugeri, D.K. Palagachev and L.G. Softova. Elliptic and parabolic equations with discontinuous coefficients. Math. Research 109, Wiley-VCH (2000).
Mclaughlin, D.W., Papanicolaou, G.C. and Pironneau, O.R., Convetion of mircrostructure and related problems. SIAM J. Appl. Math. 45 (1985) 780–797. Google Scholar
Moskow, S. and Vogelius, M., First-oder corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc. Roy. Soc. Edinburgh. 127A (1997) 1263–1299. Google Scholar
H. Owhadi and L. Zhang, Metric based up-scaling. Commun. Pure Appl. Math. LX (2007) 675–723.
Owhadi, H. and Zhang, L., Homogenization of parabolic equations with a continuum of space and time scales. SIAM J. Numer. Anal. 46 (2007) 1–36. Google Scholar
Rozza, G., Huynh, D.B.P. and Patera, A.T., 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. Google Scholar