×

Numerical methods for multiscale elliptic problems. (English) Zbl 1092.65102

This article presents an overview of the recent development on the numerical methods for elliptic problems with multiscale coefficients of the form \[ -\text{div}(a^{\varepsilon}(x)\nabla u^{\varepsilon}(x)) = f(x), \quad D, \] subject to the boundary condition \[ u^{\varepsilon}(x) = g(x), \quad \partial D, \] where \(\varepsilon \ll 1\) is a parameter that represents the ratio of the smallest and largest scales in the problem. The authors have carried out a thorough study of two representative techniques, namely, the heterogeneous multiscale method (HMM), and the multiscale finite element method (MsFEM).
An important issue is how the cost of these methods compares with techniques such as multigrid for solving the full fine scale problem. In particular, whether some special features of the problems, such as scale separation, can be exploited to save cost. Nearly all the methods reviewed do have such savings for the special problem of periodic homogenization. For more general problems, the framework of HMM still allows to take full advantage of any scale separation in the problem and therefore reduces the cost. This is difficult from the MsFEM, which in general incurs a cost that is comparable to that of solving the full scale problem.
For problems with scale separation (but without specific assumptions on the particular form of the coefficients), analytical and numerical results show that HMM gives comparable accuracy as MsFEM, with much less cost. For problems without scale separation, the numerical results suggest that HMM performs at least as well as MsFEM, in terms of accuracy and cost, even though in this case both methods may fail to converge. Since the cost of MsFEM is comparable to that of solving the full fine scale problem, one might expect that it does not need scale separation and still retains some accuracy. It is shown that this is not the case. Specifically, the authors give an example showing that if there exists an intermediate scale comparable to \(H\), the step size of the macroscale mesh, then MsFEM commits a finite error even with overlapping.

MSC:

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
65N15 Error bounds for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adams, R.A., Sobolev spaces, (1975), Academic Press · Zbl 0186.19101
[2] Alcouffe, R.E.; Brandt, A.; Dendy, J.E.; Painter, J.W., The multigrid method for the diffusion equation with strongly discontinuous coefficients, SIAM J. sci. statist. comput., 2, 430-454, (1981) · Zbl 0474.76082
[3] Allaire, G., Homogenization and two-scale convergence, SIAM J. math. anal., 23, 1482-1518, (1992) · Zbl 0770.35005
[4] Babuska, I., Homogenization and its applications, mathematical and computational problems, (), 89-116
[5] Babuska, I.; Babuska, I., Solution of interface problems by homogenization I, II, III, SIAM J. math. anal., SIAM J. math. anal., 8, 923-937, (1977), pp. 635-645 · Zbl 0402.35046
[6] Babuska, I.; Caloz, G.; Osborn, J., Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. numer. anal., 31, 945-981, (1994) · Zbl 0807.65114
[7] Babuska, I.; Osborn, J., Generalized finite element methods: their performance and their relation to mixed methods, SIAM J. numer. anal., 20, 510-536, (1983) · Zbl 0528.65046
[8] Bensoussan, A.; Lions, J.L.; Papanicolaou, G.C., Boundary layer analysis in homogenization of diffusion equations with Dirichlet cnditions on the half spaces, (), 21-40
[9] Bensoussan, A.; Lions, J.L.; Papanicolaou, G.C., Asymptotic analysis for periodic structures, (1978), North-Holland Amsterdam · Zbl 0411.60078
[10] J.F. Bourgat, Numerical experiments of the homogenization method for operators with periodic coefficients, Lecture Notes in Mathematics, vol. 707, 1977, pp. 330-356.
[11] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. comp., 31, 333-390, (1977) · Zbl 0373.65054
[12] Brezzi, F.; Franca, L.P.; Hughes, T.J.R.; Russo, A., b=∫gd{\bfx}, Comput. meth. appl. mech. eng., 145, 329-339, (1997) · Zbl 0904.76041
[13] Chen, Z.; Hou, T., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. comp., 72, 541-576, (2003) · Zbl 1017.65088
[14] Conlon, J.G.; Naddaf, A., On homogenization of elliptic equations with random coefficients, Electron J. probab., 5, 1-58, (2000) · Zbl 0956.35013
[15] Cui, J.Z.; Chao, L.Q., The two-scale analysis methods for woven composite materials, (), 203-212
[16] Durlofsky, L.J., Numerical-calculation of equivalent grid block permeability tensors for heterogenous porous media, Water resour. res., 27, 699-708, (1991)
[17] Dorobantu, M.; Engquist, B., Wavelet-based numerical homogenization, SIAM J. numer. anal., 35, 540-559, (1998) · Zbl 0936.65135
[18] E, W., Homogenization of linear and nonlinear transport equations, Comm. pure appl. math., 45, 301-326, (1992) · Zbl 0794.35014
[19] E, W.; Engquist, B., The heterogeneous multiscale methods, Commun. math. sci., 1, 87-132, (2003) · Zbl 1093.35012
[20] W. E, B. Engquist, The heterogeneous multiscale method for homogenization problems, MMS (submitted).
[21] E, W.; Engquist, B., Multiscale modeling and computation, Notice amer. math. soc., 50, 1062-1070, (2003) · Zbl 1032.65013
[22] E, W.; Ming, P.; Zhang, P., Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. am. math. soc., 18, 121-156, (2005) · Zbl 1060.65118
[23] Efendiev, Y.R.; Hou, T.; Ginting, V., Multiscale finite element methods for nonlinear problems and their applications, Commun. math. sci., 2, 553-589, (2004) · Zbl 1083.65105
[24] Efendiev, Y.R.; Hou, T.; Wu, X., Convergence of a nonconforming multiscale finite element method, SIAM J. numer. anal., 37, 888-910, (2000) · Zbl 0951.65105
[25] Engquist, B.; Runborg, O., Wavelet-based numerical homogenization with applications, (), 97-148 · Zbl 0989.65117
[26] Farhat, C.; Harari, I.; Franca, L.P., The discontinuous enrichment method, Comput. meth. appl. mech. eng., 190, 6455-6479, (2001) · Zbl 1002.76065
[27] Fish, J.; Belsky, V., Multigrid method for a periodic heterogeneous medium, part I: convergence studies for one-dimensional case, Comput. meth. appl. mech. eng., 126, 1-16, (1995) · Zbl 1067.74574
[28] Fish, J.; Belsky, V., Multigrid method for a periodic heterogeneous medium, part I: multiscale modeling and quality in multi-dimensional case, Comput. meth. appl. mech. eng., 126, 17-38, (1995) · Zbl 1067.74573
[29] Fish, J.; Yuan, Z., Multiscale enrichment based on partition of unity, Inter. J. numer. meth. eng., 62, 1341-1359, (2005) · Zbl 1078.74637
[30] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. comput. phys., 73, 325-348, (1987) · Zbl 0629.65005
[31] Hou, T.; Wu, X., A multiscale finite element method for elliptic problems in composite materials and porous media, J. comput. phys., 134, 169-189, (1997) · Zbl 0880.73065
[32] Hou, T.; Wu, X.; Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. comp., 68, 913-943, (1999) · Zbl 0922.65071
[33] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet to Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods, Comput. meth. appl. mech. eng., 127, 387-401, (1995) · Zbl 0866.76044
[34] Keller, J.B., A theorem on the conductivity of a composite medium, J. math phys., 5, 548-549, (1964) · Zbl 0129.44001
[35] Kozlov, S.M., Averaging of difference schemes, Math. USSR sbornik., 57, 351-369, (1987) · Zbl 0639.65052
[36] Matache, A.M.; Babuska, I.; Schwab, C., Generalized p-FEM in homogenization, Numer. math., 86, 319-375, (2000) · Zbl 0964.65125
[37] Matache, A.M.; Schwab, C., Two-scale finite element for homogenization problems, M2AN math. model. numer. anal., 36, 537-572, (2002) · Zbl 1070.65572
[38] Moulton, J.D.; Dendy, J.E.; Hyman, J.M., The black box multigrid numerical homogenization algorithm, J. comput. phys., 141, 1-29, (1998) · Zbl 0933.76072
[39] Neuss, N.; Jäger, W.; Wittum, G., Homoginization and multigrid, Computing, 66, 1-26, (2001) · Zbl 0992.35013
[40] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. math. anal., 20, 608-623, (1989) · Zbl 0688.35007
[41] Oden, J.T.; Vemaganti, K.S., Estimation of local modeling error and global-oriented adaptive modeling of heterogeneous materials. I: error estimates and adaptive algorithms, J. comput. phys., 164, 22-47, (2000) · Zbl 0992.74072
[42] Piatnitski, A.; Remy, E., Homogenization of elliptic difference operators, SIAM J. math. anal., 33, 53-83, (2001) · Zbl 1097.39501
[43] Sangalli, G., Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale model. simul., 1, 485-503, (2003) · Zbl 1032.65131
[44] Schwab, C., Two-scale FEM for homogenization problems, (), 91-108 · Zbl 1009.74072
[45] Schwab, C.; Matache, A.M., Generalzied FEM for homogenization problems, (), 197-238
[46] Tartar, L., Estimations de coefficients homogénéisés, Lecture notes in mathematics, vol. 704, (1979), Springer-Verlag Berlin, pp. 364-373 · Zbl 0443.35061
[47] Torquato, S., Random heterogeneous materials: microstructure and macroscopic properties, (2002), Springer-Verlag New York · Zbl 0988.74001
[48] X.Y. Yue, Numerical homogenization for well-driven transport in heterogeneus media, Report, Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, 2002.
[49] Zhikov, V.V.; Kozlov, S.M.; Oleinik, O.A., Homogenization of differential operators and integral functionals, (1994), Springer-Verlag Heidelberg · Zbl 0838.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.