A mixed multiscale finite element method for elliptic problems with oscillating coefficients. (English) Zbl 1017.65088

This paper is motivated by the numerical simulation of flow transport in highly heterogeneous porous media. By using the most elaborated tools of mathematical finite element methods, the authors analyze a mixed multiscale finite element method with an over-sampling technique which is very well suited for solving second-order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems.
By assuming that the oscillating coefficients are locally periodic and by using homogenization techniques, the authors prove the convergence of the method. Finally, they report some numerical experiments which demonstrate the efficiency and accuracy of the proposed method.
This paper is nicely written and looks like a strong mathematical and. numerical contribution to the subject.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
76M50 Homogenization applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage
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[1] T. Arbogast, Numerical subgrid upscaling of two-phase flow in porous media, TICAM Report 99-30, University of Texas at Austin, 1999. · Zbl 1072.76560
[2] Marco Avellaneda and Fang-Hua Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math. 40 (1987), no. 6, 803 – 847. · Zbl 0632.35018
[3] 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. · Zbl 0807.65114
[4] 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. · Zbl 0404.35001
[5] Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[6] F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo, \?=\int \?, Comput. Methods Appl. Mech. Engrg. 145 (1997), no. 3-4, 329 – 339. · Zbl 0904.76041
[7] Zhiming Chen, Mixed finite element methods for a dynamical Ginzburg-Landau model in superconductivity, Numer. Math. 76 (1997), no. 3, 323 – 353. · Zbl 0873.65110
[8] Zhiming Chen and Qiang Du, An upwinding mixed finite element method for a mean field model of superconducting vortices, M2AN Math. Model. Numer. Anal. 34 (2000), no. 3, 687 – 706. · Zbl 1078.82548
[9] Jim Douglas Jr. and Thomas F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19 (1982), no. 5, 871 – 885. · Zbl 0492.65051
[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] Yalchin R. Efendiev, Thomas Y. Hou, and Xiao-Hui Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37 (2000), no. 3, 888 – 910. · Zbl 0951.65105
[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] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[16] Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. · Zbl 0585.65077
[17] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[18] 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. · Zbl 0880.73065
[19] Thomas Y. Hou, Xiao-Hui Wu, and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68 (1999), no. 227, 913 – 943. · Zbl 0922.65071
[20] Усреднение дифференциал\(^{\приме}\)ных операторов, ”Наука”, Мосцощ, 1993 (Руссиан, щитх Енглиш анд Руссиан суммариес). В. В. Јиков, С. М. Козлов, анд О. А. Олейник, Хомогенизатион оф дифферентиал операторс анд интеграл фунцтионалс, Спрингер-Верлаг, Берлин, 1994. Транслатед фром тхе Руссиан бы Г. А. Ыосифиан [Г. А. Иосиф\(^{\приме}\)ян].
[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. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89 – 123. Publication No. 33. · Zbl 0341.65076
[23] Gary M. Lieberman, Oblique derivative problems in Lipschitz domains. II. Discontinuous boundary data, J. Reine Angew. Math. 389 (1988), 1 – 21. · Zbl 0648.35033
[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] Shari Moskow and Michael Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 6, 1263 – 1299. · Zbl 0888.35011
[26] O. Pironneau, On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math. 38 (1981/82), no. 3, 309 – 332. · Zbl 0505.76100
[27] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292 – 315. Lecture Notes in Math., Vol. 606.
[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. · Zbl 0572.76089
[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. · Zbl 0963.76558
[31] S. Verdière 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), no. 4, 601 – 639. · Zbl 0912.76063
[32] Li Kang Li, Discretization of the Timoshenko beam problem by the \? and the \?-\? versions of the finite element method, Numer. Math. 57 (1990), no. 4, 413 – 420. · Zbl 0683.73041
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