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

Wen-ming He; Jun-zhi Cui

doi:10.3934/cpaa.2012.11.501

Article Contents

2012, Volume 11, Issue 2: 501-516. Doi: 10.3934/cpaa.2012.11.501

This issue Previous Article Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant Next Article Nonradial positive solutions for a biharmonic critical growth problem

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

Wen-ming He^1, and
Jun-zhi Cui^2,

1.
Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang, 325035
2.
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190

Received: August 2010

Revised: April 2011

Published: March 2012

Abstract / Introduction Related Papers Cited by

Abstract

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 $\left\|G_{y}^{\varepsilon}-{G}_{y}-\theta_{G,y}^{\varepsilon}\right\|_{W^{1,q}(\Omega\ B(y,d))}$.

Keywords:

The multi-scale homogenization method,
Green's function,
Second order elliptic equations with small periodic coefficients.

Mathematics Subject Classification: 35J25.

Citation:

Related Papers

Cited by

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.