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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients
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by Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai PDF
Math. Comp. 68 (1999), 913-943 Request permission


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.
  • 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, DOI 10.1051/m2an/1991250606931
  • 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, DOI 10.1137/0731051
  • 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, DOI 10.1137/0720034
  • 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, DOI 10.1007/BFb0076258
  • 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, DOI 10.1016/0045-7825(82)90044-5
  • 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.
  • 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
  • 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
  • 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.
  • L. J. Durlofsky, Numerical-calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res. 27 (1991), 699–708.
  • 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.
  • 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, DOI 10.1002/cpa.3160430702
  • Y. R. Efendiev, Ph.D. thesis, Caltech, 1998.
  • 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, DOI 10.1137/0726016
  • J. Frehse and R. Rannacher, Eine $L^{1}$-Fehlerabschätzung für diskrete Grundlösungen in der Methode der finiten Elemente, Finite Elemente (Tagung, Univ. Bonn, Bonn, 1975) Bonn. Math. Schrift., No. 89, Inst. Angew. Math., Univ. Bonn, Bonn, 1976, pp. 92–114 (German, with English summary). MR 0471370
  • 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, DOI 10.1006/jcph.1997.5682
  • 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
  • Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
  • 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. F. Mccarthy, Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media, Transport in Porous Media 19 (1995), 123–137.
  • 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.
  • 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, DOI 10.1016/0377-0427(90)90252-U
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Additional Information
  • Thomas Y. Hou
  • Affiliation: Applied Mathematics, 217-50 California Institute of Technology Pasadena, CA 91125
  • Email:
  • 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:
  • Zhiqiang Cai
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395
  • MR Author ID: 235961
  • Email:
  • 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.
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 913-943
  • MSC (1991): Primary 65F10, 65F30
  • DOI:
  • MathSciNet review: 1642758