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

Mathematics of Computation

Volumn 72, Issue 242, 2003, Pages 541-576

  Chen, Zhiming ^a   Hou, Thomas Y ^b  

^a INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037376524     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-02-01441-2     Document Type: Article

References (32)

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, b = f∫ g, 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, RARTO 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. V.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 Functional, 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