A mixed multiscale finite element method for elliptic problems with oscillating coefficients

Authors:
Zhiming Chen and Thomas Y. Hou

Journal:
Math. Comp. **72** (2003), 541-576

MSC (2000):
Primary 65F10, 65F30

DOI:
https://doi.org/10.1090/S0025-5718-02-01441-2

Published electronically:
June 28, 2002

MathSciNet review:
1954956

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Abstract | References | Similar Articles | Additional Information

Abstract: The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is *not* required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.

**1.**T. Arbogast,*Numerical subgrid upscaling of two-phase flow in porous media,*TICAM Report 99-30, University of Texas at Austin, 1999.**2.**M. Avellaneda and F.-H. Lin,*Compactness methods in the theory of homogenization*, Comm. Pure Appl. Math.**40**(1987), 803-847. MR**88i:35019****3.**I. Babuska, G. Caloz, and E. Osborn,*Special finite element methods for a class of second order elliptic problems with rough coefficients,*SIAM J. Numer. Anal.**31**(1994), 945-981. MR**95g:65146****4.**A. Bensoussan, J.L. Lions, and G. Papanicolaou,*Asymptotic Analysis for Periodic Structures,*North-Holland, Amsterdam, 1978. MR**82h:35001****5.**F. Brezzi and M. Fortin,*Mixed and Hybrid Finite Element Methods,*Springer, New York, 1991. MR**92d:65187****6.**F. Brezzi, L.P. Franca, T.J.R. Hughes, and A. Russo, ,*Comput. Methods Appl. Mech. Engrg.***145**(1997), 329-339. MR**98g:65086****7.**Z. Chen,*Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity,*Numer. Math.**76**(1997), 323-353. MR**98d:65139****8.**Z. Chen and Q. Du,*An upwinding mixed finite element method for a mean field model of superconducting vortices,*RARIO Math. Model. Numer. Anal.**34**(2000), 687-706. MR**2001f:65133****9.**J. Douglas, Jr. and T.F. Russell,*Numerical methods for convection-dominated diffusion problem based on combining the method of characteristics with finite element or finite difference procedures,*SIAM J. Numer. Anal.**19**(1982), 871-885. MR**84b:65093****10.**L.J. Durlofsky,*Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media,*Water Resources Research**27**(1991), 699-708.**11.**L.J. Durlofsky, R.C. Jones, and W.J. Milliken,*A nonuniform coarsening approach for the scale-up of displacement processes in heterogeneous porous media,*Adv. Water Resources,**20**(1997), 335-347.**12.**Y.R. Efendiev,*The Multiscale Finite Element Method and its Applications,*Ph.D. thesis, California Institute of Technology, 1999.**13.**Y.R. Efendiev, T.Y. Hou, and X.H. Wu,*The convergence of nonconforming multiscale finite element methods,*SIAM J. Numer. Anal.**37**(2000), 888-910. MR**2002a:65176****14.**Y.R. Efendiev, L.J. Durlofsky, and S.H. Lee,*Modeling of subgrid effects in coarse scale simulations of transport in heterogeneous porous media,*Water Resources Research**36**(2000), 2031-2041.**15.**D. Gilbarg and N.S. Trudinger,*Elliptic Partial Differential Equations of Second Order,*Springer-Verlag, Berlin, 1983. MR**86c:35035****16.**V. Girault and P.-A. Raviart,*Finite Element Methods for Navier-Stokes Equations,*Springer, Berlin, 1986. MR**88b:65129****17.**P. Grisvard,*Elliptic Problems on Nonsmooth Domains,*Pitman, Boston, 1985. MR**86m:35044****18.**T.Y. Hou and X.H. Wu,*A multiscale finite element method for elliptic problems in composite materials and porous media,*J. Comput. Phys.**134**(1997), 169-189. MR**98e:73132****19.**T.Y. Hou, X.H. Wu, and Z. Cai,*Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,*Math. Comp.**68**(1999), 913-943. MR**99i:65126****20.**V.V. Jikov, S.M. Kozlov, and O.A. Oleinik,*Homogenization of Differential Operators and Integral Functionals,*Springer, Berlin, 1994. MR**96h:35003b****21.**P. Langlo and M.S. Espedal,*Macrodispersion for two-phase, immiscible flow in porous media,*Adv. Water Resources**17**(1994), 297-316.**22.**P. Lesaint and P.-A. Raviart, ``On a Finite Element Method for Solving the Neutron Transport equation'', in*Mathematical Aspects of the Finite Element Method in Partial Differential Equations*, ed. by C. de Boor, Academic Press, New York, 1974. MR**58:31918****23.**G.M. Lieberman,*Oblique derivative problems in Lipschitz domains II. Discontinuous boundary data,*J. Reine Angew. Math.**389**(1988), 1-21. MR**89h:35094****24.**J.F. McCarthy,*Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media,*Transport in Porous Media,**19**(1995), 123-137.**25.**S. Moskow and M. Vogelius,*First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof,*Proc. Royal Soc. Edinburgh A**127**(1997), 1263-1299. MR**99g:35018****26.**O. Pironneau,*On the transport-diffusion algorithm and its application to the Navier-Stokes equations*, Numer. Math.**38**(1982), 309-332. MR**83d:65258****27.**P. Raviart and J. Thomas, ``A mixed element method for 2nd order elliptic problems'', in*Mathematical Aspects of the Finite Element Method*, Lecture Notes on Mathematics 606, Springer, Berlin, 1977. MR**58:3547****28.**T.F. Russell and M.F. Wheeler, ``Finite element and finite difference methods for continuous flows in porous media'', in*The Mathematics of Reservoir Simulation,*R.E. Ewing, ed., SIAM, Philadelphia, 1983.**29.**L. Tartar, ``Nonlocal Effect Induced by Homogenization'', in*PDEs and Calculus of Variations*, F. Columbini, ed., Birkhäuser Publ., Boston, 1989.**30.**T.C. Wallstrom, S.L. Hou, M.A. Christie, L.J. Durlofsky and D.H. Sharp,*Accurate scale up of two phase flow using renormalization and nonuniform coarsening,*Computational Geosciences**3**(1999), 69-87.**31.**S. Verdiere and M.H. Vignal,*Numerical and theoretical study of a dual mesh method using finite volume schemes for two-phase flow problems in porous media,*Numer. Math.**80**(1998), 601-639. MR**99g:76117****32.**P.M. De Zeeuw,*Matrix-dependent prolongation and restrictions in a blackbox multigrid solver,*J. Comput. Appl. Math.**33**(1990), 1-27. MR**92e:65152**

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Additional Information

**Zhiming Chen**

Affiliation:
LSEC, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China

Email:
zmchen@lsec.cc.ac.cn

**Thomas Y. Hou**

Affiliation:
Applied Mathematics, California Institute of Technology, Pasadena, California 91125.

Email:
hou@acm.caltech.edu

DOI:
https://doi.org/10.1090/S0025-5718-02-01441-2

Received by editor(s):
March 21, 2000

Received by editor(s) in revised form:
July 10, 2000, and May 29, 2001

Published electronically:
June 28, 2002

Additional Notes:
The first author was supported in part by China NSF under the grants 19771080 and 10025102 and by China MOS under the grant G1999032804.

The second author was supported in part by NSF under the grant DMS-0073916 and by ARO under the grant DAAD19-99-1-0141.

Article copyright:
© Copyright 2002
American Mathematical Society