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Pointwise estimate for elliptic equations in periodic perforated domains
1. | Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 30050, Taiwan |
References:
[1] |
E. Acerbi, V. Chiado Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, 18 (1992), 481-496.
doi: 10.1016/0362-546X(92)90015-7. |
[2] | |
[3] |
Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[4] |
R. C. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media. I: An integral representation and its consequences, SIAM J. Math. Anal., 22 (1991), 1-15.
doi: 10.1137/0522001. |
[5] |
N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials, Kluwer Academic Publishers, 1989.
doi: 10.1007/978-94-009-2247-1. |
[6] |
B. Berkowitz, et al., Physical pictures of transport in heterogeneous media: Advection-dispersion, random walk, and fractional derivative formulations, Water resources Research, 38 (2002), 1191-1194. |
[7] |
Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer, 2008.
doi: 10.1007/978-0-387-75934-0. |
[8] |
Alain Bensoussan, Jacques-Louis Lions and George Papanicolaou, Asymptotic Analysis for Periodic Structures, Elsevier North-Holland, 1978. |
[9] |
Li-Qun Cao, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains, Numerische Mathematik, 103 (2006), 11-45.
doi: 10.1007/s00211-005-0668-4. |
[10] |
Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978. |
[11] |
D. Cioranescu, A. Damlamian, G. Grisoa and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008), 248-277.
doi: 10.1016/j.matpur.2007.12.008. |
[12] |
M. Giaquinta, Multiple integrals in the calculus of variations, Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second edition, 1983.
doi: 10.1007/978-3-642-61798-0. |
[14] |
Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), 913-943.
doi: 10.1090/S0025-5718-99-01077-7. |
[15] |
V.V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-Verlag, 1994.
doi: 10.1007/978-3-642-84659-5. |
[16] |
Viviane Klein and Malgorzata Peszynska, Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model, Journal of Applied Mathematics, (2012) (2012), 26 pages. |
[17] |
J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, The Rocky Mountain J. Math., 10 (1980), 125-144.
doi: 10.1216/RMJ-1980-10-1-125. |
[18] |
N. Neuss, W. Jäger and G. Wittum, Homogenization and multigrid, Computing, 66 (2001), 1-26.
doi: 10.1007/s006070170036. |
[19] |
O. A. Oleinik, A. S. Shamaev and G. A. Tosifan, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992. |
show all references
References:
[1] |
E. Acerbi, V. Chiado Piat, G. Dal Maso and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis, 18 (1992), 481-496.
doi: 10.1016/0362-546X(92)90015-7. |
[2] | |
[3] |
Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[4] |
R. C. Morgan and I. Babuska, An approach for constructing families of homogenized equations for periodic media. I: An integral representation and its consequences, SIAM J. Math. Anal., 22 (1991), 1-15.
doi: 10.1137/0522001. |
[5] |
N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials, Kluwer Academic Publishers, 1989.
doi: 10.1007/978-94-009-2247-1. |
[6] |
B. Berkowitz, et al., Physical pictures of transport in heterogeneous media: Advection-dispersion, random walk, and fractional derivative formulations, Water resources Research, 38 (2002), 1191-1194. |
[7] |
Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer, 2008.
doi: 10.1007/978-0-387-75934-0. |
[8] |
Alain Bensoussan, Jacques-Louis Lions and George Papanicolaou, Asymptotic Analysis for Periodic Structures, Elsevier North-Holland, 1978. |
[9] |
Li-Qun Cao, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains, Numerische Mathematik, 103 (2006), 11-45.
doi: 10.1007/s00211-005-0668-4. |
[10] |
Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, Amsterdam: North-Holland, 1978. |
[11] |
D. Cioranescu, A. Damlamian, G. Grisoa and D. Onofrei, The periodic unfolding method for perforated domains and Neumann sieve models, J. Math. Pures Appl., 89 (2008), 248-277.
doi: 10.1016/j.matpur.2007.12.008. |
[12] |
M. Giaquinta, Multiple integrals in the calculus of variations, Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983. |
[13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, second edition, 1983.
doi: 10.1007/978-3-642-61798-0. |
[14] |
Thomas Y. Hou, Xiao-Hui Wu and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 68 (1999), 913-943.
doi: 10.1090/S0025-5718-99-01077-7. |
[15] |
V.V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-Verlag, 1994.
doi: 10.1007/978-3-642-84659-5. |
[16] |
Viviane Klein and Malgorzata Peszynska, Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model, Journal of Applied Mathematics, (2012) (2012), 26 pages. |
[17] |
J. L. Lions, Asymptotic expansions in perforated media with a periodic structure, The Rocky Mountain J. Math., 10 (1980), 125-144.
doi: 10.1216/RMJ-1980-10-1-125. |
[18] |
N. Neuss, W. Jäger and G. Wittum, Homogenization and multigrid, Computing, 66 (2001), 1-26.
doi: 10.1007/s006070170036. |
[19] |
O. A. Oleinik, A. S. Shamaev and G. A. Tosifan, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992. |
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