March  2012, 11(2): 501-516. doi: 10.3934/cpaa.2012.11.501

The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$

1. 

Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, 325035, China

2. 

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received  August 2010 Revised  April 2011 Published  October 2011

In this paper, for the second order elliptic problems with small periodic coefficients of the form $\frac{\partial}{\partial x_{i}} (a^{i j}(\frac{x}{\varepsilon})\frac{\partial u^{\varepsilon}(x)}{\partial x_{j}})=f(x)$, we shall discuss the multi-scale homogenization theory for Green's function $G_{y}^{\varepsilon}$ at point $y\in\Omega$ on Sobolev space $W^{1,q}(\Omega)$. Assume that $B(y,d)=\{x\in\Omega|dist(x,y)\leq d\},$ ${G}_{y}$ and $\theta_{G,y}^{\varepsilon}$ are the 1-order approximation and the boundary corrector of $G_{y}^{\varepsilon}$, respectively. We present an estimate for $\left\|G_{y}^{\varepsilon}-{G}_{y}-\theta_{G,y}^{\varepsilon}\right\|_{W^{1,q}(\Omega\ B(y,d))}$.
Citation: Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure and Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
References:
[1]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization," Amsterdam, North-Holland, 1992.

[2]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer, Berlin Heidelberg New York, 1994.

[3]

S. M. Kozolev, Asymptotic of fundamental solutions of second-order divergence differential equations, Math. USSR Sbornik, 41 (1982), 249-267.

[4]

N. S. Bachvalov and G. P.Panasenko, "Homogenization of Processes in Periodic Media," Moscow, Nauka, 1984.

[5]

E. Sanchez-Palencia, "Nonhomogeneous Media and Vibration Theory," Lect. Notes Phys., 127, Springer, Berlin Heidelberg New York, 1980.

[6]

A. Azzam, Smoothness properties of bounded solution of Dirichlet problem for elliptic equations in regions with corners on the boundaries, Canad. Math. Bull., 23 (1980).

[7]

M. Avellaneda and F. H. Lin, Theory of homogenization, Comm. Pure Appl. Math, 42 (1989), 803-847.

[8]

D. Cioranescu, J. Saint and J. Paulin, "Homogenization of Reticulated Structures," Springer, Berlin, Heidelberg New York, 1998.

[9]

U. Hornung, "Homogenization and Porous Media," Springer, Berlin Heidelberg New York, 1996.

[10]

A. Bensussan, J. L. Lions and G. Papanicolou, "Asymptotic Analysis of Periodic Structures," North-Holland, Amsterdan, 1978.

[11]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. doi: DOI:10.1137/0523084.

[12]

W. M. He and J. Z. Cui, A pointwise estimate on the 1-order approximation of $G^{\varepsilon}_{x_0}$, IMA Journal of Applied Mathematics, 70 (2005), 241-269. doi: DOI:10.1093/imamat/hxh029.

[13]

T. Y. Hou, Convergence of a multi-scale finite element method for elliptic problem with oscillation coefficients, Math. Comp., 68 (1999), 913-943.

[14]

Z. M. Chen, W. B. Deng and H. Ye, A new upscaling method for the solute transport equations, Discrete and Continuous Dynamical Systems-series B, 13 (2005), 493-515.

[15]

Z. M. Chen and T. Y. Hou, A mixed multi-scale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2002), 541-576. doi: DOI:10.1090/S0025-5718-02-01441-2.

[16]

Z. M. Chen and X. Y. Yue, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media, SIAM Multiscale Model. Simul., 1 (2003), 260-303. doi: DOI:10.1137/S1540345902413322.

[17]

R. Efrndiev Yalchin, T. Y. Hou and X. H. Wu, Convergence of nonconforming multi-scale finite element method, SIAM J. Numer. Anal., 37 (2000), 888-910. doi: DOI:10.1137/S0036142997330329.

[18]

P. B. Ming and X. Y. Yue, Numerical methods for multiscale elliptic problems, Journal of Computational Physics, 214 (2006), 421-445. doi: DOI:10.1016/j.jcp.2005.09.024.

[19]

W. E, P. B. Ming and P. W. Zhang, Analysis of the heterogeneous multi-scale method for elliptic homogenization problems, Journal of the American Mathematical Society, 18 (2005), 121-156.

[20]

O. A. Ladyzhenskaia and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," New York, Academic Press, 1968.

[21]

L. Q. Cao and J. Z. Cui, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problems for second order elliptic equations in perforated domains, Numer. Math., 96 (2004), 525-581. doi: DOI:10.1007/s00211-003-0468-7.

[22]

L. Q. Cao, Multi-scale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains, Numer. Math., 103 (2005), 11-45. doi: DOI:10.1007/s00211-005-0668-4.

[23]

X. Wang and L. Q. Cao, The hole-filling method and the uniform multi-scale computation of the elastic equations in perforated domains, International Journal of Numerical Analysis and Modeling, 5 (2008), 612-634.

[24]

W. M. He and J. Z. Cui, A finite element method for elliptic problems with rapidly oscillating coefficients, BIT Numerical Mathematics, 47 (2007), 77-102. doi: DOI: 10.1007/s10543-007-0117-0.

show all references

References:
[1]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization," Amsterdam, North-Holland, 1992.

[2]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals," Springer, Berlin Heidelberg New York, 1994.

[3]

S. M. Kozolev, Asymptotic of fundamental solutions of second-order divergence differential equations, Math. USSR Sbornik, 41 (1982), 249-267.

[4]

N. S. Bachvalov and G. P.Panasenko, "Homogenization of Processes in Periodic Media," Moscow, Nauka, 1984.

[5]

E. Sanchez-Palencia, "Nonhomogeneous Media and Vibration Theory," Lect. Notes Phys., 127, Springer, Berlin Heidelberg New York, 1980.

[6]

A. Azzam, Smoothness properties of bounded solution of Dirichlet problem for elliptic equations in regions with corners on the boundaries, Canad. Math. Bull., 23 (1980).

[7]

M. Avellaneda and F. H. Lin, Theory of homogenization, Comm. Pure Appl. Math, 42 (1989), 803-847.

[8]

D. Cioranescu, J. Saint and J. Paulin, "Homogenization of Reticulated Structures," Springer, Berlin, Heidelberg New York, 1998.

[9]

U. Hornung, "Homogenization and Porous Media," Springer, Berlin Heidelberg New York, 1996.

[10]

A. Bensussan, J. L. Lions and G. Papanicolou, "Asymptotic Analysis of Periodic Structures," North-Holland, Amsterdan, 1978.

[11]

G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518. doi: DOI:10.1137/0523084.

[12]

W. M. He and J. Z. Cui, A pointwise estimate on the 1-order approximation of $G^{\varepsilon}_{x_0}$, IMA Journal of Applied Mathematics, 70 (2005), 241-269. doi: DOI:10.1093/imamat/hxh029.

[13]

T. Y. Hou, Convergence of a multi-scale finite element method for elliptic problem with oscillation coefficients, Math. Comp., 68 (1999), 913-943.

[14]

Z. M. Chen, W. B. Deng and H. Ye, A new upscaling method for the solute transport equations, Discrete and Continuous Dynamical Systems-series B, 13 (2005), 493-515.

[15]

Z. M. Chen and T. Y. Hou, A mixed multi-scale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2002), 541-576. doi: DOI:10.1090/S0025-5718-02-01441-2.

[16]

Z. M. Chen and X. Y. Yue, Numerical homogenization of well singularities in the flow transport through heterogeneous porous media, SIAM Multiscale Model. Simul., 1 (2003), 260-303. doi: DOI:10.1137/S1540345902413322.

[17]

R. Efrndiev Yalchin, T. Y. Hou and X. H. Wu, Convergence of nonconforming multi-scale finite element method, SIAM J. Numer. Anal., 37 (2000), 888-910. doi: DOI:10.1137/S0036142997330329.

[18]

P. B. Ming and X. Y. Yue, Numerical methods for multiscale elliptic problems, Journal of Computational Physics, 214 (2006), 421-445. doi: DOI:10.1016/j.jcp.2005.09.024.

[19]

W. E, P. B. Ming and P. W. Zhang, Analysis of the heterogeneous multi-scale method for elliptic homogenization problems, Journal of the American Mathematical Society, 18 (2005), 121-156.

[20]

O. A. Ladyzhenskaia and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," New York, Academic Press, 1968.

[21]

L. Q. Cao and J. Z. Cui, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problems for second order elliptic equations in perforated domains, Numer. Math., 96 (2004), 525-581. doi: DOI:10.1007/s00211-003-0468-7.

[22]

L. Q. Cao, Multi-scale asymptotic expansion and finite element methods for the mixed boundary value problems of second order elliptic equation in perforated domains, Numer. Math., 103 (2005), 11-45. doi: DOI:10.1007/s00211-005-0668-4.

[23]

X. Wang and L. Q. Cao, The hole-filling method and the uniform multi-scale computation of the elastic equations in perforated domains, International Journal of Numerical Analysis and Modeling, 5 (2008), 612-634.

[24]

W. M. He and J. Z. Cui, A finite element method for elliptic problems with rapidly oscillating coefficients, BIT Numerical Mathematics, 47 (2007), 77-102. doi: DOI: 10.1007/s10543-007-0117-0.

[1]

Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure and Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164

[2]

Thomas Y. Hou, Pengfei Liu. Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4451-4476. doi: 10.3934/dcds.2016.36.4451

[3]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[4]

Emiliano Cristiani, Elisa Iacomini. An interface-free multi-scale multi-order model for traffic flow. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6189-6207. doi: 10.3934/dcdsb.2019135

[5]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449

[6]

Eugene Kashdan, Svetlana Bunimovich-Mendrazitsky. Multi-scale model of bladder cancer development. Conference Publications, 2011, 2011 (Special) : 803-812. doi: 10.3934/proc.2011.2011.803

[7]

Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200

[8]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure and Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[9]

Thomas Blanc, Mihaï Bostan. Multi-scale analysis for highly anisotropic parabolic problems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 335-399. doi: 10.3934/dcdsb.2019186

[10]

Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013

[11]

Hongjie Dong, Seick Kim. Green's functions for parabolic systems of second order in time-varying domains. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1407-1433. doi: 10.3934/cpaa.2014.13.1407

[12]

Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375

[13]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

[14]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[15]

Tudor Barbu. Deep learning-based multiple moving vehicle detection and tracking using a nonlinear fourth-order reaction-diffusion based multi-scale video object analysis. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022083

[16]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[17]

Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451

[18]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139

[19]

Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543

[20]

Jean-Philippe Bernard, Emmanuel Frénod, Antoine Rousseau. Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects. Conference Publications, 2013, 2013 (special) : 69-76. doi: 10.3934/proc.2013.2013.69

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]