# American Institute of Mathematical Sciences

September  2012, 7(3): 503-524. doi: 10.3934/nhm.2012.7.503

## Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems

 1 Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster

Received  November 2011 Revised  May 2012 Published  October 2012

In this work, we are concerned with the convergence of the multiscale finite element method (MsFEM) for elliptic homogenization problems, where we do not assume a certain periodic or stochastic structure, but an averaging assumption which in particular covers periodic and ergodic stochastic coefficients. We also give a result on the convergence in the case of an arbitrary coupling between grid size $H$ and a parameter $\epsilon$. $\epsilon$ is an indicator for the size of the fine scale which converges to zero. The findings of this work are based on the homogenization results obtained in [B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (4), 2011].
Citation: Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks and Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503
##### References:
 [1] J. E. Aarnes and Y. Efendiev, Mixed multiscale finite element methods for stochastic porous media flows, SIAM J. Sci. Comput., 30 (2008), 2319-2339. doi: 10.1137/07070108X. [2] J. E. Aarnes, Y. Efendiev and L. Jiang, Mixed multiscale finite element methods using limited global information, Multiscale Model. Simul., 7 (2008), 655-676. [3] A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model. Simul., 4 (2005), 447-459 (electronic). [4] A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems, J. Comput. Phys., 191 (2003), 18-39. doi: 10.1016/S0021-9991(03)00303-6. [5] M. Bourlard, M. Dauge, M. S. Lubuma and S. Nicaise, Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III. Finite element methods on polygonal domains, SIAM J. Numer. Anal., 29 (1992), 136-155. doi: 10.1137/0729009. [6] Z. Chen, M. Cui, T. Savchuk and X. Yu, The multiscale finite element method with nonconforming elements for elliptic homogenization problems, Multiscale Model. Simul., 7 (2008), 517-538. [7] Z. Chen and T. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems,, SIAM J. Numer. Anal., 46 (): 260. [8] C. C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), 1915-1955. doi: 10.1090/S0025-5718-2010-02372-5. [9] D. Cioranescu and P. Donato, "An Introduction to Homogenization," The Clarendon Press Oxford University Press, New York, 1999. [10] P. Dostert, Y. Efendiev and T. Y. Hou, Multiscale finite element methods for stochastic porous media flow equations and applications to uncertainty quantification, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3445-3455. doi: 10.1016/j.cma.2008.02.030. [11] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), 87-132. [12] W. E and B. Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc., 50 (2003), 1062-1070. [13] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156 (electronic). doi: 10.1090/S0894-0347-04-00469-2. [14] Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications, Appl. Numer. Math., 57 (2007), 577-596. doi: 10.1016/j.apnum.2006.07.009. [15] Y. Efendiev and T. Hou, "Multiscale Finite Element Methods," Surveys and Tutorials in the Applied Mathematical Sciences, 4, Springer, New York, 2009. [16] Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Commun. Math. Sci., 2 (2004), 553-589. [17] Y. Efendiev and A. Pankov, Numerical homogenization of monotone elliptic operators, Multiscale Model. Simul., 2 (2003), 62-79. [18] Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations, SIAM J. Appl. Math., 65 (2004), 43-68. doi: 10.1137/S0036139903424886. [19] Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators, Multiscale Model. Simul., 2 (2004), 237-268. [20] Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Adv. Differential Equations, 10 (2005), 1235-1260. [21] Y. Efendiev and A. Pankov, On homogenization of almost periodic nonlinear parabolic operators, Int. J. Evol. Equ., 1 (2005), 203-209. [22] Y. Efendiev, J. Galvis and X. H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys., 230 (2011), 937-955. doi: 10.1016/j.jcp.2010.09.026. [23] Y. Efendiev, T. Y. Hou and X. H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), 888-910. doi: 10.1137/S0036142997330329. [24] P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [25] P. Grisvard, "Singularities in Boundary Value Problems," Recherches en Mathématiques Appliquées, 22, Masson, Paris, 1992. [26] P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains, Numer. Math., 113 (2009), 601-629. doi: 10.1007/s00211-009-0244-4. [27] P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Netw. Heterog. Media, 5 (2010), 711-744. [28] V. H. Hoang, Sparse finite element method for periodic multiscale nonlinear monotone problems, Multiscale Model. Simul., 7 (2008), 1042-1072. [29] V. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales,, Multiscale Model. Simul., 3 (): 168.  doi: 10.1137/030601077. [30] U. Hornung, "Homogenization and Porous Media," Interdisciplinary Applied Mathematics, 6, Springer, New York, 1997. [31] T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), 169-189. doi: 10.1006/jcph.1997.5682. [32] T. Y. Hou, X. H. Wu and Z. 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. [33] T. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387-401. doi: 10.1016/0045-7825(95)00844-9. [34] T. Hughes, G. R. Feijóo, L. Mazzei and J. B. Quincy, The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3-24. doi: 10.1016/S0045-7825(98)00079-6. [35] M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: duality techniques for elliptic problems, in "Multiscale Methods in Science and Engineering," Lect. Notes Comput. Sci. Eng., Springer, Berlin, 44 (2005), 181-193. [36] M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2313-2324. doi: 10.1016/j.cma.2006.08.019. [37] M. G. Larson and A. Målqvist, An adaptive variational multiscale method for convection-diffusion problems, Comm. Numer. Methods Engrg., 25 (2009), 65-79. doi: 10.1002/cnm.1106. [38] M. G. Larson and A. Målqvist, A mixed adaptive variational multiscale method with applications in oil reservoir simulation, Math. Models Methods Appl. Sci., 19 (2009), 1017-1042. doi: 10.1142/S021820250900370X. [39] J. Li, A multiscale finite element method for optimal control problems governed by the elliptic homogenization equations, Comput. Math. Appl., 60 (2010), 390-398. doi: 10.1016/j.camwa.2010.04.017. [40] A. M. Matache, Sparse two-scale FEM for homogenization problems, Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala) J. Sci. Comput., 17 (2002), 659-669. [41] A. M. Matache and C. Schwab, Two-scale FEM for homogenization problems, M2AN Math. Model. Numer. Anal., 36 (2002), 537-572. doi: 10.1051/m2an:2002025. [42] J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems, Multiscale Model. Simul., 7 (2008), 171-196. [43] J. M. Nordbotten, Adaptive variational multiscale methods for multiphase flow in porous media, Multiscale Model. Simul., 7 (2008), 1455-1473. [44] M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Model. Simul., 4 (2005), 88-114 (electronic). [45] C. Schwab and A.-M. Matache, Generalized {FEM for homogenization problems}, in "Multiscale and Multiresolution Methods," vol 20 of 'Lect. Notes Comput. Sci. Eng.', Springer, Berlin (2002), 197-237. [46] B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (2011), 755-781. [47] H. W. Zhang, J. K. Wu, J. Lü and Z. D. Fu, Extended multiscale finite element method for mechanical analysis of heterogeneous materials, Acta Mech. Sin., 26 (2010), 899-920. doi: 10.1007/s10409-010-0393-9.

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##### References:
 [1] J. E. Aarnes and Y. Efendiev, Mixed multiscale finite element methods for stochastic porous media flows, SIAM J. Sci. Comput., 30 (2008), 2319-2339. doi: 10.1137/07070108X. [2] J. E. Aarnes, Y. Efendiev and L. Jiang, Mixed multiscale finite element methods using limited global information, Multiscale Model. Simul., 7 (2008), 655-676. [3] A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model. Simul., 4 (2005), 447-459 (electronic). [4] A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems, J. Comput. Phys., 191 (2003), 18-39. doi: 10.1016/S0021-9991(03)00303-6. [5] M. Bourlard, M. Dauge, M. S. Lubuma and S. Nicaise, Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III. Finite element methods on polygonal domains, SIAM J. Numer. Anal., 29 (1992), 136-155. doi: 10.1137/0729009. [6] Z. Chen, M. Cui, T. Savchuk and X. Yu, The multiscale finite element method with nonconforming elements for elliptic homogenization problems, Multiscale Model. Simul., 7 (2008), 517-538. [7] Z. Chen and T. Savchuk, Analysis of the multiscale finite element method for nonlinear and random homogenization problems,, SIAM J. Numer. Anal., 46 (): 260. [8] C. C. Chu, I. G. Graham and T. Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), 1915-1955. doi: 10.1090/S0025-5718-2010-02372-5. [9] D. Cioranescu and P. Donato, "An Introduction to Homogenization," The Clarendon Press Oxford University Press, New York, 1999. [10] P. Dostert, Y. Efendiev and T. Y. Hou, Multiscale finite element methods for stochastic porous media flow equations and applications to uncertainty quantification, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3445-3455. doi: 10.1016/j.cma.2008.02.030. [11] W. E and B. Engquist, The heterogeneous multiscale methods, Commun. Math. Sci., 1 (2003), 87-132. [12] W. E and B. Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc., 50 (2003), 1062-1070. [13] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156 (electronic). doi: 10.1090/S0894-0347-04-00469-2. [14] Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications, Appl. Numer. Math., 57 (2007), 577-596. doi: 10.1016/j.apnum.2006.07.009. [15] Y. Efendiev and T. Hou, "Multiscale Finite Element Methods," Surveys and Tutorials in the Applied Mathematical Sciences, 4, Springer, New York, 2009. [16] Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications, Commun. Math. Sci., 2 (2004), 553-589. [17] Y. Efendiev and A. Pankov, Numerical homogenization of monotone elliptic operators, Multiscale Model. Simul., 2 (2003), 62-79. [18] Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations, SIAM J. Appl. Math., 65 (2004), 43-68. doi: 10.1137/S0036139903424886. [19] Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators, Multiscale Model. Simul., 2 (2004), 237-268. [20] Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators, Adv. Differential Equations, 10 (2005), 1235-1260. [21] Y. Efendiev and A. Pankov, On homogenization of almost periodic nonlinear parabolic operators, Int. J. Evol. Equ., 1 (2005), 203-209. [22] Y. Efendiev, J. Galvis and X. H. Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys., 230 (2011), 937-955. doi: 10.1016/j.jcp.2010.09.026. [23] Y. Efendiev, T. Y. Hou and X. H. Wu, Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., 37 (2000), 888-910. doi: 10.1137/S0036142997330329. [24] P. Grisvard, "Elliptic Problems in Nonsmooth Domains," Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. [25] P. Grisvard, "Singularities in Boundary Value Problems," Recherches en Mathématiques Appliquées, 22, Masson, Paris, 1992. [26] P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains, Numer. Math., 113 (2009), 601-629. doi: 10.1007/s00211-009-0244-4. [27] P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift, Netw. Heterog. Media, 5 (2010), 711-744. [28] V. H. Hoang, Sparse finite element method for periodic multiscale nonlinear monotone problems, Multiscale Model. Simul., 7 (2008), 1042-1072. [29] V. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales,, Multiscale Model. Simul., 3 (): 168.  doi: 10.1137/030601077. [30] U. Hornung, "Homogenization and Porous Media," Interdisciplinary Applied Mathematics, 6, Springer, New York, 1997. [31] T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134 (1997), 169-189. doi: 10.1006/jcph.1997.5682. [32] T. Y. Hou, X. H. Wu and Z. 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. [33] T. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387-401. doi: 10.1016/0045-7825(95)00844-9. [34] T. Hughes, G. R. Feijóo, L. Mazzei and J. B. Quincy, The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3-24. doi: 10.1016/S0045-7825(98)00079-6. [35] M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: duality techniques for elliptic problems, in "Multiscale Methods in Science and Engineering," Lect. Notes Comput. Sci. Eng., Springer, Berlin, 44 (2005), 181-193. [36] M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2313-2324. doi: 10.1016/j.cma.2006.08.019. [37] M. G. Larson and A. Målqvist, An adaptive variational multiscale method for convection-diffusion problems, Comm. Numer. Methods Engrg., 25 (2009), 65-79. doi: 10.1002/cnm.1106. [38] M. G. Larson and A. Målqvist, A mixed adaptive variational multiscale method with applications in oil reservoir simulation, Math. Models Methods Appl. Sci., 19 (2009), 1017-1042. doi: 10.1142/S021820250900370X. [39] J. Li, A multiscale finite element method for optimal control problems governed by the elliptic homogenization equations, Comput. Math. Appl., 60 (2010), 390-398. doi: 10.1016/j.camwa.2010.04.017. [40] A. M. Matache, Sparse two-scale FEM for homogenization problems, Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala) J. Sci. Comput., 17 (2002), 659-669. [41] A. M. Matache and C. Schwab, Two-scale FEM for homogenization problems, M2AN Math. Model. Numer. Anal., 36 (2002), 537-572. doi: 10.1051/m2an:2002025. [42] J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems, Multiscale Model. Simul., 7 (2008), 171-196. [43] J. M. Nordbotten, Adaptive variational multiscale methods for multiphase flow in porous media, Multiscale Model. Simul., 7 (2008), 1455-1473. [44] M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Model. Simul., 4 (2005), 88-114 (electronic). [45] C. Schwab and A.-M. Matache, Generalized {FEM for homogenization problems}, in "Multiscale and Multiresolution Methods," vol 20 of 'Lect. Notes Comput. Sci. Eng.', Springer, Berlin (2002), 197-237. [46] B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (2011), 755-781. [47] H. W. Zhang, J. K. Wu, J. Lü and Z. D. Fu, Extended multiscale finite element method for mechanical analysis of heterogeneous materials, Acta Mech. Sin., 26 (2010), 899-920. doi: 10.1007/s10409-010-0393-9.
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