Article Contents
Article Contents

# Deterministic homogenization for media with barriers

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

 Citation:

•  [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.