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Optimal local multi-scale basis functions for linear elliptic equations with rough coefficients
1. | 1200 E California Blvd, MC 9-94, California Institute of Technology, Pasadena, CA 91125, United States, United States |
References:
[1] |
G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[2] |
G. Allaire, Shape optimization by the homogenization method, vol. 146, Springer Science & Business Media, 2012.
doi: 10.1007/978-1-4684-9286-6. |
[3] |
G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Modeling & Simulation, 4 (2005), 790-812.
doi: 10.1137/040611239. |
[4] |
I. Babuška, G. Caloz and J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM Journal on Numerical Analysis, 31 (1994), 945-981.
doi: 10.1137/0731051. |
[5] |
I. Babuška, X. Huang and R. Lipton, Machine computation using the exponentially convergent multiscale spectral generalized finite element method, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 493-515.
doi: 10.1051/m2an/2013117. |
[6] |
I. Babuska and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Modeling & Simulation, 9 (2011), 373-406.
doi: 10.1137/100791051. |
[7] |
I. Babuška and J. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods, SIAM Journal on Numerical Analysis, 20 (1983), 510-536.
doi: 10.1137/0720034. |
[8] |
I. Babuška and J. Osborn, Can a finite element method perform arbitrarily badly?, Mathematics of Computation of the American Mathematical Society, 69 (2000), 443-462.
doi: 10.1090/S0025-5718-99-01085-6. |
[9] |
A. Bensoussan, J. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374, American Mathematical Soc., 2011. |
[10] |
L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for rational mechanics and analysis, 198 (2010), 677-721.
doi: 10.1007/s00205-010-0302-1. |
[11] |
Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Mathematics of Computation, 72 (2003), 541-576.
doi: 10.1090/S0025-5718-02-01441-2. |
[12] |
C. Chu, I. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Mathematics of Computation, 79 (2010), 1915-1955.
doi: 10.1090/S0025-5718-2010-02372-5. |
[13] |
J. Chu, Y. Efendiev, V. Ginting and T. Y. Hou, Flow based oversampling technique for multiscale finite element methods, Advances in Water Resources, 31 (2008), 599-608.
doi: 10.1016/j.advwatres.2007.11.005. |
[14] |
M. Ci, T. Y. Hou and Z. Shi, A multiscale model reduction method for partial differential equations, ESAIM-Mathematical Modelling and Numerical Analysis, 48 (2014), 449-474.
doi: 10.1051/m2an/2013115. |
[15] |
D. Cioranescu and P. Donato, Introduction to, homogenization., ().
|
[16] |
E. De Giorgi, New problems in $\gamma$-convergence and g-convergence, Free boundary problems, 2 (1980), 183-194. |
[17] |
E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area,, Ennio De Giorgi, ().
|
[18] |
Y. Efendiev, V. Ginting, T. Y. Hou and R. Ewing, Accurate multiscale finite element methods for two-phase flow simulations, Journal of Computational Physics, 220 (2006), 155-174.
doi: 10.1016/j.jcp.2006.05.015. |
[19] |
Y. Efendiev and T. Y. Hou, Multiscale finite element methods for porous media flows and their applications, Applied Numerical Mathematics, 57 (2007), 577-596.
doi: 10.1016/j.apnum.2006.07.009. |
[20] |
Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods: Theory and Applications, vol. 4, Springer Science & Business Media, 2009. |
[21] |
Y. Efendiev, T. Y. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Communications in Mathematical Sciences, 2 (2004), 553-589.
doi: 10.4310/CMS.2004.v2.n4.a2. |
[22] |
Y. Efendiev, T. Y. Hou and X. Wu, Convergence of a nonconforming multiscale finite element method, SIAM Journal on Numerical Analysis, 37 (2000), 888-910.
doi: 10.1137/S0036142997330329. |
[23] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer Science & Business Media, 2001. |
[24] |
A. Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies, Multiscale Modeling & Simulation, 5 (2006), 996-1043.
doi: 10.1137/060649112. |
[25] |
P. Henning and D. Peterseim, Oversampling for the multiscale finite element method, Multiscale Modeling & Simulation, 11 (2013), 1149-1175.
doi: 10.1137/120900332. |
[26] |
T. Y. Hou and P. Liu, A heterogeneous stochastic fem framework for elliptic pdes, Journal of Computational Physics, 281 (2015), 942-969.
doi: 10.1016/j.jcp.2014.10.020. |
[27] |
T. Y. Hou, P. Liu and Z. Zhang, A model reduction method for elliptic pdes with random input using the heterogeneous stochastic fem framework, Bulletin of the Institute of Mathematics, 11 (2016), 179-216. |
[28] |
T. Y. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of computational physics, 134 (1997), 169-189.
doi: 10.1006/jcph.1997.5682. |
[29] |
T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation of the American Mathematical Society, 68 (1999), 913-943.
doi: 10.1090/S0025-5718-99-01077-7. |
[30] |
T. Y. Hou, X. Wu and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a petrov-galerkin formulation, Communications in Mathematical Sciences, 2 (2004), 185-205.
doi: 10.4310/CMS.2004.v2.n2.a3. |
[31] |
P. Jenny, S. Lee and H. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, Journal of Computational Physics, 187 (2003), 47-67.
doi: 10.1016/S0021-9991(03)00075-5. |
[32] |
P. Jenny, S. Lee and H. Tchelepi, Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Modeling & Simulation, 3 (2005), 50-64.
doi: 10.1137/030600795. |
[33] |
V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Science & Business Media, 2012.
doi: 10.1007/978-3-642-84659-5. |
[34] |
S. Kozlov, Averaging of random operators, Matematicheskii Sbornik, 151 (1979), 188-202. |
[35] |
I. Lunati and P. Jenny, Multiscale finite-volume method for compressible multiphase flow in porous media, Journal of Computational Physics, 216 (2006), 616-636.
doi: 10.1016/j.jcp.2006.01.001. |
[36] |
A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Mathematics of Computation, 83 (2014), 2583-2603.
doi: 10.1090/S0025-5718-2014-02868-8. |
[37] |
J. Melenk, On n-widths for elliptic problems, Journal of mathematical analysis and applications, 247 (2000), 272-289.
doi: 10.1006/jmaa.2000.6862. |
[38] |
J. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications, Computer methods in applied mechanics and engineering, 139 (1996), 289-314.
doi: 10.1016/S0045-7825(96)01087-0. |
[39] |
R. Millward, A New Adaptive Multiscale Finite Element Method with Applications to High Contrast Interface Problems, PhD thesis, University of Bath, 2011. |
[40] |
F. Murat, Compacité par compensation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 5 (1978), 489-507. |
[41] | |
[42] |
H. Owhadi, Bayesian numerical homogenization, Multiscale Modeling & Simulation, 13 (2015), 812-828.
doi: 10.1137/140974596. |
[43] |
H. Owhadi, Mult-grid with rough coefficients and multiresolution operator decomposition from hierarchical information games,, arXiv preprint arXiv:1503.03467., ().
|
[44] |
H. Owhadi and L. Zhang, Metric-based upscaling, Communications on Pure and Applied Mathematics, 60 (2007), 675-723.
doi: 10.1002/cpa.20163. |
[45] |
H. Owhadi, L. Zhang and L. Berlyand, Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 517-552.
doi: 10.1051/m2an/2013118. |
[46] |
N. Panasenko and N. Bakhvalov, Homogenization: Averaging processes in periodic media: Mathematical problems in the mechanics of composite materials,, 1989., ().
doi: 10.1007/978-94-009-2247-1. |
[47] |
D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors,, , ().
|
[48] |
A. Pinkus, $n$-Width in Approximation Theory, Springer, 1985.
doi: 10.1007/978-3-642-69894-1. |
[49] |
S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 22 (1968), 571-597. |
[50] |
S. Spagnolo, Convergence in energy for elliptic operators,, Numerical Solutions of Partial Differential Equations III, ().
|
show all references
References:
[1] |
G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis, 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[2] |
G. Allaire, Shape optimization by the homogenization method, vol. 146, Springer Science & Business Media, 2012.
doi: 10.1007/978-1-4684-9286-6. |
[3] |
G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, Multiscale Modeling & Simulation, 4 (2005), 790-812.
doi: 10.1137/040611239. |
[4] |
I. Babuška, G. Caloz and J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM Journal on Numerical Analysis, 31 (1994), 945-981.
doi: 10.1137/0731051. |
[5] |
I. Babuška, X. Huang and R. Lipton, Machine computation using the exponentially convergent multiscale spectral generalized finite element method, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 493-515.
doi: 10.1051/m2an/2013117. |
[6] |
I. Babuska and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Modeling & Simulation, 9 (2011), 373-406.
doi: 10.1137/100791051. |
[7] |
I. Babuška and J. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods, SIAM Journal on Numerical Analysis, 20 (1983), 510-536.
doi: 10.1137/0720034. |
[8] |
I. Babuška and J. Osborn, Can a finite element method perform arbitrarily badly?, Mathematics of Computation of the American Mathematical Society, 69 (2000), 443-462.
doi: 10.1090/S0025-5718-99-01085-6. |
[9] |
A. Bensoussan, J. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 374, American Mathematical Soc., 2011. |
[10] |
L. Berlyand and H. Owhadi, Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Archive for rational mechanics and analysis, 198 (2010), 677-721.
doi: 10.1007/s00205-010-0302-1. |
[11] |
Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Mathematics of Computation, 72 (2003), 541-576.
doi: 10.1090/S0025-5718-02-01441-2. |
[12] |
C. Chu, I. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Mathematics of Computation, 79 (2010), 1915-1955.
doi: 10.1090/S0025-5718-2010-02372-5. |
[13] |
J. Chu, Y. Efendiev, V. Ginting and T. Y. Hou, Flow based oversampling technique for multiscale finite element methods, Advances in Water Resources, 31 (2008), 599-608.
doi: 10.1016/j.advwatres.2007.11.005. |
[14] |
M. Ci, T. Y. Hou and Z. Shi, A multiscale model reduction method for partial differential equations, ESAIM-Mathematical Modelling and Numerical Analysis, 48 (2014), 449-474.
doi: 10.1051/m2an/2013115. |
[15] |
D. Cioranescu and P. Donato, Introduction to, homogenization., ().
|
[16] |
E. De Giorgi, New problems in $\gamma$-convergence and g-convergence, Free boundary problems, 2 (1980), 183-194. |
[17] |
E. De Giorgi, Sulla convergenza di alcune successioni d'integrali del tipo dell'area,, Ennio De Giorgi, ().
|
[18] |
Y. Efendiev, V. Ginting, T. Y. Hou and R. Ewing, Accurate multiscale finite element methods for two-phase flow simulations, Journal of Computational Physics, 220 (2006), 155-174.
doi: 10.1016/j.jcp.2006.05.015. |
[19] |
Y. Efendiev and T. Y. Hou, Multiscale finite element methods for porous media flows and their applications, Applied Numerical Mathematics, 57 (2007), 577-596.
doi: 10.1016/j.apnum.2006.07.009. |
[20] |
Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods: Theory and Applications, vol. 4, Springer Science & Business Media, 2009. |
[21] |
Y. Efendiev, T. Y. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Communications in Mathematical Sciences, 2 (2004), 553-589.
doi: 10.4310/CMS.2004.v2.n4.a2. |
[22] |
Y. Efendiev, T. Y. Hou and X. Wu, Convergence of a nonconforming multiscale finite element method, SIAM Journal on Numerical Analysis, 37 (2000), 888-910.
doi: 10.1137/S0036142997330329. |
[23] |
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224, Springer Science & Business Media, 2001. |
[24] |
A. Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies, Multiscale Modeling & Simulation, 5 (2006), 996-1043.
doi: 10.1137/060649112. |
[25] |
P. Henning and D. Peterseim, Oversampling for the multiscale finite element method, Multiscale Modeling & Simulation, 11 (2013), 1149-1175.
doi: 10.1137/120900332. |
[26] |
T. Y. Hou and P. Liu, A heterogeneous stochastic fem framework for elliptic pdes, Journal of Computational Physics, 281 (2015), 942-969.
doi: 10.1016/j.jcp.2014.10.020. |
[27] |
T. Y. Hou, P. Liu and Z. Zhang, A model reduction method for elliptic pdes with random input using the heterogeneous stochastic fem framework, Bulletin of the Institute of Mathematics, 11 (2016), 179-216. |
[28] |
T. Y. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of computational physics, 134 (1997), 169-189.
doi: 10.1006/jcph.1997.5682. |
[29] |
T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Mathematics of Computation of the American Mathematical Society, 68 (1999), 913-943.
doi: 10.1090/S0025-5718-99-01077-7. |
[30] |
T. Y. Hou, X. Wu and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a petrov-galerkin formulation, Communications in Mathematical Sciences, 2 (2004), 185-205.
doi: 10.4310/CMS.2004.v2.n2.a3. |
[31] |
P. Jenny, S. Lee and H. Tchelepi, Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, Journal of Computational Physics, 187 (2003), 47-67.
doi: 10.1016/S0021-9991(03)00075-5. |
[32] |
P. Jenny, S. Lee and H. Tchelepi, Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Modeling & Simulation, 3 (2005), 50-64.
doi: 10.1137/030600795. |
[33] |
V. Jikov, S. Kozlov and O. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Science & Business Media, 2012.
doi: 10.1007/978-3-642-84659-5. |
[34] |
S. Kozlov, Averaging of random operators, Matematicheskii Sbornik, 151 (1979), 188-202. |
[35] |
I. Lunati and P. Jenny, Multiscale finite-volume method for compressible multiphase flow in porous media, Journal of Computational Physics, 216 (2006), 616-636.
doi: 10.1016/j.jcp.2006.01.001. |
[36] |
A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Mathematics of Computation, 83 (2014), 2583-2603.
doi: 10.1090/S0025-5718-2014-02868-8. |
[37] |
J. Melenk, On n-widths for elliptic problems, Journal of mathematical analysis and applications, 247 (2000), 272-289.
doi: 10.1006/jmaa.2000.6862. |
[38] |
J. Melenk and I. Babuška, The partition of unity finite element method: Basic theory and applications, Computer methods in applied mechanics and engineering, 139 (1996), 289-314.
doi: 10.1016/S0045-7825(96)01087-0. |
[39] |
R. Millward, A New Adaptive Multiscale Finite Element Method with Applications to High Contrast Interface Problems, PhD thesis, University of Bath, 2011. |
[40] |
F. Murat, Compacité par compensation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 5 (1978), 489-507. |
[41] | |
[42] |
H. Owhadi, Bayesian numerical homogenization, Multiscale Modeling & Simulation, 13 (2015), 812-828.
doi: 10.1137/140974596. |
[43] |
H. Owhadi, Mult-grid with rough coefficients and multiresolution operator decomposition from hierarchical information games,, arXiv preprint arXiv:1503.03467., ().
|
[44] |
H. Owhadi and L. Zhang, Metric-based upscaling, Communications on Pure and Applied Mathematics, 60 (2007), 675-723.
doi: 10.1002/cpa.20163. |
[45] |
H. Owhadi, L. Zhang and L. Berlyand, Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 517-552.
doi: 10.1051/m2an/2013118. |
[46] |
N. Panasenko and N. Bakhvalov, Homogenization: Averaging processes in periodic media: Mathematical problems in the mechanics of composite materials,, 1989., ().
doi: 10.1007/978-94-009-2247-1. |
[47] |
D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors,, , ().
|
[48] |
A. Pinkus, $n$-Width in Approximation Theory, Springer, 1985.
doi: 10.1007/978-3-642-69894-1. |
[49] |
S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 22 (1968), 571-597. |
[50] |
S. Spagnolo, Convergence in energy for elliptic operators,, Numerical Solutions of Partial Differential Equations III, ().
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