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February  2015, 8(1): 29-44. doi: 10.3934/dcdss.2015.8.29

Deterministic homogenization for media with barriers

Vsevolod Laptev 1,

1. 

Saratovskaya 9, 160, Moscow, 109518, Russian Federation

Received  February 2013 Revised  June 2013 Published  July 2014

Averaging coefficient in a second order elliptic equation is a well known and important model problem. Additional to non-periodic rapid oscillations, the coefficient may contain barriers and channels - long and narrow bodies with low or high values of the coefficient. When the length of such structures is comparable with the problem size - there is no scale separation.
    In this article we consider coefficients with barriers. We show how the averaged coefficient may be inadequate near the barriers and propose a generalization which can detect the potential problems and improve the accuracy of the averaged solution.
Keywords: homogenization, elliptic, averaging, two-scale extensions, Non-periodic, numerical., upscaling, coarse graining.
Mathematics Subject Classification: 35B27, 35B40, 35J25, 80M40, 76M50, 65N1.
Citation: Vsevolod Laptev. Deterministic homogenization for media with barriers. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 29-44. doi: 10.3934/dcdss.2015.8.29
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978.  Google Scholar

[3]

Y. Capdeville and J. J. Marigo, Second order homogenization of the elastic wave equation for non-periodic layered media, Geophysical Journal International, 170 (2007), 823-838. doi: 10.1111/j.1365-246X.2007.03462.x.  Google Scholar

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Y. Chen, L. J. Durlofsky, M. Gerritsen and X. H. Wen, A coupled local-global upscaling ap- proach for simulating flow in highly heterogeneous formations, Advances in Water Resources, 26 (2003), 1041-1060. doi: 10.1016/S0309-1708(03)00101-5.  Google Scholar

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C. C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comput., 79 (2010), 1915-1955. doi: 10.1090/S0025-5718-2010-02372-5.  Google Scholar

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L. J. Durlofsky, Numerical calculation of equivalent gridblock permeability tensors for heterogeneous porous media, Water Resources Research, 27 (1991), 699-708. doi: 10.1029/91WR00107.  Google Scholar

[7]

L. J. Durlofsky, Upscaling and gridding of fine scale geological models for flow simulation, Proceedings of the 8th International Forum on Reservoir Simulation in Stresa, Italy, 2005, 59 pp. Google Scholar

[8]

Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys., 251 (2013), 116-135. doi: 10.1016/j.jcp.2013.04.045.  Google Scholar

[9]

C. L. Farmer, Upscaling: A review, Numerical Methods in Fluids, 40 (2002), 63-78. doi: 10.1002/fld.267.  Google Scholar

[10]

H. Hajibeygi, G. Bonfigli, M. A. Hesse and P. Jenny, Iterative multiscale finite-volume method, Journal of Computational Physics, 227 (2008), 8604-8621. doi: 10.1016/j.jcp.2008.06.013.  Google Scholar

[11]

T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134 (1997), 169-189. doi: 10.1006/jcph.1997.5682.  Google Scholar

[12]

V. Laptev and S. Belouettar, On averaging of the non-periodic conductivity coefficient using two-scale extension, PAMM, 5 (2005), 681-682. doi: 10.1002/pamm.200510316.  Google Scholar

[13]

V. Laptev, Two-scale extensions for non-periodic coefficients,, preprint, ().   Google Scholar

[14]

V. Laptev, On numerical averaging of the conductivity coefficient using two-scale extensions,, preprint, ().   Google Scholar

[15]

V. D. Laptev, Construction and practical use of two-scaled extensions for rapidly oscillating functions, Journal of Mathematical Sciences, 158 (2009), 211-218. doi: 10.1007/s10958-009-9384-4.  Google Scholar

[16]

V. Laptev, work, in progress., ().   Google Scholar

[17]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[18]

H. Owhadi and L. Zhang, Metric based up-scaling,, preprint, ().   Google Scholar

[19]

H. Owhadi and L. Zhang, Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast, Multiscale Model. Simul., 9 (2011), 1373-1398. doi: 10.1137/100813968.  Google Scholar

[20]

X. H. Wen, L. J. Durlofsky and M. G. Edwards, Use of border regions for improved permeability upscaling, Mathematical Geology, 35 (2003), 521-547. doi: 10.1023/A:1026230617943.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978.  Google Scholar

[3]

Y. Capdeville and J. J. Marigo, Second order homogenization of the elastic wave equation for non-periodic layered media, Geophysical Journal International, 170 (2007), 823-838. doi: 10.1111/j.1365-246X.2007.03462.x.  Google Scholar

[4]

Y. Chen, L. J. Durlofsky, M. Gerritsen and X. H. Wen, A coupled local-global upscaling ap- proach for simulating flow in highly heterogeneous formations, Advances in Water Resources, 26 (2003), 1041-1060. doi: 10.1016/S0309-1708(03)00101-5.  Google Scholar

[5]

C. C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comput., 79 (2010), 1915-1955. doi: 10.1090/S0025-5718-2010-02372-5.  Google Scholar

[6]

L. J. Durlofsky, Numerical calculation of equivalent gridblock permeability tensors for heterogeneous porous media, Water Resources Research, 27 (1991), 699-708. doi: 10.1029/91WR00107.  Google Scholar

[7]

L. J. Durlofsky, Upscaling and gridding of fine scale geological models for flow simulation, Proceedings of the 8th International Forum on Reservoir Simulation in Stresa, Italy, 2005, 59 pp. Google Scholar

[8]

Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys., 251 (2013), 116-135. doi: 10.1016/j.jcp.2013.04.045.  Google Scholar

[9]

C. L. Farmer, Upscaling: A review, Numerical Methods in Fluids, 40 (2002), 63-78. doi: 10.1002/fld.267.  Google Scholar

[10]

H. Hajibeygi, G. Bonfigli, M. A. Hesse and P. Jenny, Iterative multiscale finite-volume method, Journal of Computational Physics, 227 (2008), 8604-8621. doi: 10.1016/j.jcp.2008.06.013.  Google Scholar

[11]

T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of Computational Physics, 134 (1997), 169-189. doi: 10.1006/jcph.1997.5682.  Google Scholar

[12]

V. Laptev and S. Belouettar, On averaging of the non-periodic conductivity coefficient using two-scale extension, PAMM, 5 (2005), 681-682. doi: 10.1002/pamm.200510316.  Google Scholar

[13]

V. Laptev, Two-scale extensions for non-periodic coefficients,, preprint, ().   Google Scholar

[14]

V. Laptev, On numerical averaging of the conductivity coefficient using two-scale extensions,, preprint, ().   Google Scholar

[15]

V. D. Laptev, Construction and practical use of two-scaled extensions for rapidly oscillating functions, Journal of Mathematical Sciences, 158 (2009), 211-218. doi: 10.1007/s10958-009-9384-4.  Google Scholar

[16]

V. Laptev, work, in progress., ().   Google Scholar

[17]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis, 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[18]

H. Owhadi and L. Zhang, Metric based up-scaling,, preprint, ().   Google Scholar

[19]

H. Owhadi and L. Zhang, Localized bases for finite dimensional homogenization approximations with non-separated scales and high-contrast, Multiscale Model. Simul., 9 (2011), 1373-1398. doi: 10.1137/100813968.  Google Scholar

[20]

X. H. Wen, L. J. Durlofsky and M. G. Edwards, Use of border regions for improved permeability upscaling, Mathematical Geology, 35 (2003), 521-547. doi: 10.1023/A:1026230617943.  Google Scholar

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