# American Institute of Mathematical Sciences

August  2016, 36(8): 4451-4476. doi: 10.3934/dcds.2016.36.4451

## 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

Received  June 2015 Revised  October 2015 Published  March 2016

This paper addresses a multi-scale finite element method for second order linear elliptic equations with rough coefficients, which is based on the compactness of the solution operator, and does not depend on any scale-separation or periodicity assumption of the coefficient. We consider a special type of basis functions, the multi-scale basis, which are harmonic on each element and show that they have optimal approximation property for fixed local boundary conditions. To build the optimal local boundary conditions, we introduce a set of interpolation basis functions, and reduce our problem to approximating the interpolation residual of the solution space on each edge of the coarse mesh. And this is achieved through the singular value decompositions of some local oversampling operators. Rigorous error control can be obtained through thresholding in constructing the basis functions. The optimal interpolation basis functions are also identified and they can be constructed by solving some local least square problems. Numerical results for several problems with rough coefficients and high contrast inclusions are presented to demonstrate the capacity of our method in identifying and exploiting the compact structure of the local solution space to achieve computational savings.
Citation: 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
##### 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, 414. [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] F. Murat and L. Tartar, H-convergence, Springer, 1997. [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, arXiv:1505.07611. [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, Acad. Press, New York.

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##### 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, 414. [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] F. Murat and L. Tartar, H-convergence, Springer, 1997. [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, arXiv:1505.07611. [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, Acad. Press, New York.
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