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 & 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, (1992). Google Scholar

[2]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,", Springer, (1994). Google Scholar

[3]

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

[4]

N. S. Bachvalov and G. P.Panasenko, "Homogenization of Processes in Periodic Media,", Moscow, (1984). Google Scholar

[5]

E. Sanchez-Palencia, "Nonhomogeneous Media and Vibration Theory,", Lect. Notes Phys., (1980). Google Scholar

[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). Google Scholar

[7]

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

[8]

D. Cioranescu, J. Saint and J. Paulin, "Homogenization of Reticulated Structures,", Springer, (1998). Google Scholar

[9]

U. Hornung, "Homogenization and Porous Media,", Springer, (1996). Google Scholar

[10]

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

[11]

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

[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. doi: DOI:10.1093/imamat/hxh029. Google Scholar

[13]

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

[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. Google Scholar

[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. doi: DOI:10.1090/S0025-5718-02-01441-2. Google Scholar

[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. doi: DOI:10.1137/S1540345902413322. Google Scholar

[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. doi: DOI:10.1137/S0036142997330329. Google Scholar

[18]

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

[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. Google Scholar

[20]

O. A. Ladyzhenskaia and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", New York, (1968). Google Scholar

[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. doi: DOI:10.1007/s00211-003-0468-7. Google Scholar

[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. doi: DOI:10.1007/s00211-005-0668-4. Google Scholar

[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. Google Scholar

[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. doi: DOI: 10.1007/s10543-007-0117-0. Google Scholar

show all references

References:
[1]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, "Mathematical Problems in Elasticity and Homogenization,", Amsterdam, (1992). Google Scholar

[2]

V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,", Springer, (1994). Google Scholar

[3]

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

[4]

N. S. Bachvalov and G. P.Panasenko, "Homogenization of Processes in Periodic Media,", Moscow, (1984). Google Scholar

[5]

E. Sanchez-Palencia, "Nonhomogeneous Media and Vibration Theory,", Lect. Notes Phys., (1980). Google Scholar

[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). Google Scholar

[7]

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

[8]

D. Cioranescu, J. Saint and J. Paulin, "Homogenization of Reticulated Structures,", Springer, (1998). Google Scholar

[9]

U. Hornung, "Homogenization and Porous Media,", Springer, (1996). Google Scholar

[10]

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

[11]

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

[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. doi: DOI:10.1093/imamat/hxh029. Google Scholar

[13]

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

[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. Google Scholar

[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. doi: DOI:10.1090/S0025-5718-02-01441-2. Google Scholar

[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. doi: DOI:10.1137/S1540345902413322. Google Scholar

[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. doi: DOI:10.1137/S0036142997330329. Google Scholar

[18]

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

[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. Google Scholar

[20]

O. A. Ladyzhenskaia and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", New York, (1968). Google Scholar

[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. doi: DOI:10.1007/s00211-003-0468-7. Google Scholar

[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. doi: DOI:10.1007/s00211-005-0668-4. Google Scholar

[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. Google Scholar

[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. doi: DOI: 10.1007/s10543-007-0117-0. Google Scholar

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