American Institute of Mathematical Sciences

Advanced
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.  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

show all references

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

[1]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[2]

Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks & Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503
[3]

Ben Schweizer, Marco Veneroni. The needle problem approach to non-periodic homogenization. Networks & Heterogeneous Media, 2011, 6 (4) : 755-781. doi: 10.3934/nhm.2011.6.755
[4]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[5]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[6]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[7]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143
[8]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[9]

Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827
[10]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[11]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[12]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
[13]

Katherine A. Bold, Karthikeyan Rajendran, Balázs Ráth, Ioannis G. Kevrekidis. An equation-free approach to coarse-graining the dynamics of networks. Journal of Computational Dynamics, 2014, 1 (1) : 111-134. doi: 10.3934/jcd.2014.1.111
[14]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[15]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65
[16]

Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061
[17]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall. Networks & Heterogeneous Media, 2008, 3 (3) : 651-673. doi: 10.3934/nhm.2008.3.651
[18]

Yao Xu, Weisheng Niu. Periodic homogenization of elliptic systems with stratified structure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2295-2323. doi: 10.3934/dcds.2019097
[19]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[20]

Patrizia Donato, Florian Gaveau. Homogenization and correctors for the wave equation in non periodic perforated domains. Networks & Heterogeneous Media, 2008, 3 (1) : 97-124. doi: 10.3934/nhm.2008.3.97

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences