Deterministic homogenization for media with barriers

Vsevolod Laptev

doi:10.3934/dcdss.2015.8.29

Article Contents

2015, Volume 8, Issue 1: 29-44. Doi: 10.3934/dcdss.2015.8.29

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Deterministic homogenization for media with barriers

Vsevolod Laptev^1,

1.
Saratovskaya 9, 160, Moscow, 109518

Received: January 31, 2013

Revised: May 31, 2013

Published: January 2015

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Abstract

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:

Mathematics Subject Classification: 35B27, 35B40, 35J25, 80M40, 76M50, 65N12.

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References

[1]	G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.doi: 10.1137/0523084.
[2]	A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structure, North Holland, Amsterdam, 1978.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	C. L. Farmer, Upscaling: A review, Numerical Methods in Fluids, 40 (2002), 63-78.doi: 10.1002/fld.267.
[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.
[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.
[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.
[13]	V. Laptev, Two-scale extensions for non-periodic coefficients, preprint, arXiv:math/0512123.
[14]	V. Laptev, On numerical averaging of the conductivity coefficient using two-scale extensions, preprint, arXiv:0710.2072.
[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.
[16]	V. Laptev, work in progress.
[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.
[18]	H. Owhadi and L. Zhang, Metric based up-scaling, preprint, arXiv:math/0505223.
[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.
[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.