Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

Authors:
Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai

Journal:
Math. Comp. **68** (1999), 913-943

MSC (1991):
Primary 65F10, 65F30

Published electronically:
March 3, 1999

MathSciNet review:
1642758

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are *adapted* to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is *not* required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

**1.**M. Avellaneda, Th. Y. Hou, and G. C. Papanicolaou,*Finite difference approximations for partial differential equations with rapidly oscillating coefficients*, RAIRO Modél. Math. Anal. Numér.**25**(1991), no. 6, 693–710 (English, with French summary). MR**1135990****2.**Ivo Babuška, Gabriel Caloz, and John E. Osborn,*Special finite element methods for a class of second order elliptic problems with rough coefficients*, SIAM J. Numer. Anal.**31**(1994), no. 4, 945–981. MR**1286212**, 10.1137/0731051**3.**I. Babuška and J. E. Osborn,*Generalized finite element methods: their performance and their relation to mixed methods*, SIAM J. Numer. Anal.**20**(1983), no. 3, 510–536. MR**701094**, 10.1137/0720034**4.**I. Babuška and J. E. Osborn,*Finite element methods for the solution of problems with rough input data*, Singularities and constructive methods for their treatment (Oberwolfach, 1983) Lecture Notes in Math., vol. 1121, Springer, Berlin, 1985, pp. 1–18. MR**806382**, 10.1007/BFb0076258**5.**I. Babuška and W. G. Szymczak,*An error analysis for the finite element method applied to convection diffusion problems*, Comput. Methods Appl. Mech. Engrg.**31**(1982), no. 1, 19–42. MR**669258**, 10.1016/0045-7825(82)90044-5**6.**J. Bear,*Use of models in decision making*, Transport and Reactive Processes in Aquifers (T. H. Dracos and F. Stauffer, eds.), Balkema, Rotterdam, 1994, pp. 3-9.**7.**A. Bensoussan, J.-L. Lions, and G. Papanicolaou,*Boundary layer analysis in homogeneization of diffusion equations with Dirichlet conditions in the half space*, Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) Wiley, New York-Chichester-Brisbane, 1978, pp. 21–40. MR**536001****8.**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. MR**503330****9.**M. E. Cruz and A. Petera,*A parallel Monte-Carlo finite-element procedure for the analysis of multicomponent random media*, Int. J. Numer. Methods Eng.**38**(1995), 1087-1121.**10.**L. J. Durlofsky,*Numerical-calculation of equivalent grid block permeability tensors for heterogeneous porous media*, Water Resour. Res.**27**(1991), 699-708.**11.**B. B. Dykaar and P. K. Kitanidis,*Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 1. Method*, Water Resour. Res.**28**(1992), 1155-1166.**12.**Weinan E and Thomas Y. Hou,*Homogenization and convergence of the vortex method for 2-D Euler equations with oscillatory vorticity fields*, Comm. Pure Appl. Math.**43**(1990), no. 7, 821–855. MR**1072394**, 10.1002/cpa.3160430702**13.**Y. R. Efendiev, Ph.D. thesis, Caltech, 1998.**14.**Björn Engquist and Thomas Y. Hou,*Particle method approximation of oscillatory solutions to hyperbolic differential equations*, SIAM J. Numer. Anal.**26**(1989), no. 2, 289–319. MR**987391**, 10.1137/0726016**15.**J. Frehse and R. Rannacher,*Eine 𝐿¹-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente*, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 92–114. Bonn. Math. Schrift., No. 89 (German, with English summary). MR**0471370****16.**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. MR**1455261**, 10.1006/jcph.1997.5682**17.**S. M. Kozlov,*Averaging of differential operators with almost periodic rapidly oscillating coefficients*, Mat. Sb. (N.S.)**107(149)**(1978), no. 2, 199–217, 317 (Russian). MR**512007****18.**Olga A. Ladyzhenskaya and Nina N. Ural′tseva,*Linear and quasilinear elliptic equations*, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR**0244627****19.**J.-L. Lions and E. Magenes,*Problèmes aux limites non homogènes et applications. Vol. 1*, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR**0247243**

J.-L. Lions and E. Magenes,*Non-homogeneous boundary value problems and applications. Vol. I*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR**0350177****20.**J. F. Mccarthy,*Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media*, Transport in Porous Media**19**(1995), 123-137.**21.**S. Moskow and M. Vogelius,*First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof*, Proc. Roy. Soc. Edinburgh Sect. A.**127**(1997), 1263-1299. CMP**98:06****22.**P. M. de Zeeuw,*Matrix-dependent prolongations and restrictions in a blackbox multigrid solver*, J. Comput. Appl. Math.**33**(1990), no. 1, 1–27. MR**1081238**, 10.1016/0377-0427(90)90252-U

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65F10,
65F30

Retrieve articles in all journals with MSC (1991): 65F10, 65F30

Additional Information

**Thomas Y. Hou**

Affiliation:
Applied Mathematics, 217-50 California Institute of Technology Pasadena, CA 91125

Email:
hou@ama.caltech.edu

**Xiao-Hui Wu**

Affiliation:
Applied Mathematics, 217-50, California Institute of Technology, Pasadena, CA 91125

Address at time of publication:
Exxon Production Research Company, P. O. Box 2189, Houston, TX 77252

Email:
xwu@ama.caltech.edu

**Zhiqiang Cai**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395

Email:
zcai@math.purdue.edu

DOI:
https://doi.org/10.1090/S0025-5718-99-01077-7

Keywords:
Multiscale base functions,
finite element,
homogenization,
oscillating coefficients

Received by editor(s):
August 5, 1996

Received by editor(s) in revised form:
November 10, 1997

Published electronically:
March 3, 1999

Additional Notes:
This work is supported in part by ONR under the grant N00014-94-0310, by DOE under the grant DE-FG03-89ER25073, and by NSF under the grant DMS-9704976.

Article copyright:
© Copyright 1999
American Mathematical Society