Deterministic homogenization for media with barriers

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

Deterministic homogenization for media with barriers

1.	Saratovskaya 9, 160, Moscow, 109518, Russian Federation

Received February 2013 Revised June 2013 Published July 2014

Abstract
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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. doi: 10.1137/0523084. Google Scholar
[2]	A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure,, North Holland, (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. 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. 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. 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. 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, (2005). Google Scholar
[8]	Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods (GMsFEM),, J. Comput. Phys., 251* (2013), 116. doi: 10.1016/j.jcp.2013.04.045. Google Scholar*
[9]	C. L. Farmer, Upscaling: A review,, Numerical Methods in Fluids, 40 (2002), 63. 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. 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. 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. 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. 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. 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. 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. doi: 10.1023/A:1026230617943. Google Scholar

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References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[2]	A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure,, North Holland, (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. 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. 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. 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. 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, (2005). Google Scholar
[8]	Y. Efendiev, J. Galvis and T. Hou, Generalized multiscale finite element methods (GMsFEM),, J. Comput. Phys., 251* (2013), 116. doi: 10.1016/j.jcp.2013.04.045. Google Scholar*
[9]	C. L. Farmer, Upscaling: A review,, Numerical Methods in Fluids, 40 (2002), 63. 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. 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. 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. 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. 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. 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. 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. doi: 10.1023/A:1026230617943. Google Scholar

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