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 & 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. doi: 10.1137/07070108X. Google Scholar

[2]

J. E. Aarnes, Y. Efendiev and L. Jiang, Mixed multiscale finite element methods using limited global information,, Multiscale Model. Simul., 7 (2008), 655. Google Scholar

[3]

A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM,, Multiscale Model. Simul., 4 (2005), 447. Google Scholar

[4]

A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems,, J. Comput. Phys., 191 (2003), 18. doi: 10.1016/S0021-9991(03)00303-6. Google Scholar

[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. doi: 10.1137/0729009. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1090/S0025-5718-2010-02372-5. Google Scholar

[9]

D. Cioranescu and P. Donato, "An Introduction to Homogenization,", The Clarendon Press Oxford University Press, (1999). Google Scholar

[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. doi: 10.1016/j.cma.2008.02.030. Google Scholar

[11]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. Google Scholar

[12]

W. E and B. Engquist, Multiscale modeling and computation,, Notices Amer. Math. Soc., 50 (2003), 1062. Google Scholar

[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. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[14]

Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications,, Appl. Numer. Math., 57 (2007), 577. doi: 10.1016/j.apnum.2006.07.009. Google Scholar

[15]

Y. Efendiev and T. Hou, "Multiscale Finite Element Methods,", Surveys and Tutorials in the Applied Mathematical Sciences, 4 (2009). Google Scholar

[16]

Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications,, Commun. Math. Sci., 2 (2004), 553. Google Scholar

[17]

Y. Efendiev and A. Pankov, Numerical homogenization of monotone elliptic operators,, Multiscale Model. Simul., 2 (2003), 62. Google Scholar

[18]

Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations,, SIAM J. Appl. Math., 65 (2004), 43. doi: 10.1137/S0036139903424886. Google Scholar

[19]

Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators,, Multiscale Model. Simul., 2 (2004), 237. Google Scholar

[20]

Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators,, Adv. Differential Equations, 10 (2005), 1235. Google Scholar

[21]

Y. Efendiev and A. Pankov, On homogenization of almost periodic nonlinear parabolic operators,, Int. J. Evol. Equ., 1 (2005), 203. Google Scholar

[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. doi: 10.1016/j.jcp.2010.09.026. Google Scholar

[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. doi: 10.1137/S0036142997330329. Google Scholar

[24]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 24 (1985). Google Scholar

[25]

P. Grisvard, "Singularities in Boundary Value Problems,", Recherches en Mathématiques Appliquées, 22 (1992). Google Scholar

[26]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numer. Math., 113 (2009), 601. doi: 10.1007/s00211-009-0244-4. Google Scholar

[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. Google Scholar

[28]

V. H. Hoang, Sparse finite element method for periodic multiscale nonlinear monotone problems,, Multiscale Model. Simul., 7 (2008), 1042. Google Scholar

[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. Google Scholar

[30]

U. Hornung, "Homogenization and Porous Media,", Interdisciplinary Applied Mathematics, 6 (1997). Google Scholar

[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. doi: 10.1006/jcph.1997.5682. Google Scholar

[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. doi: 10.1090/S0025-5718-99-01077-7. Google Scholar

[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. doi: 10.1016/0045-7825(95)00844-9. Google Scholar

[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. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[35]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: duality techniques for elliptic problems,, in, 44 (2005), 181. Google Scholar

[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. doi: 10.1016/j.cma.2006.08.019. Google Scholar

[37]

M. G. Larson and A. Målqvist, An adaptive variational multiscale method for convection-diffusion problems,, Comm. Numer. Methods Engrg., 25 (2009), 65. doi: 10.1002/cnm.1106. Google Scholar

[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. doi: 10.1142/S021820250900370X. Google Scholar

[39]

J. Li, A multiscale finite element method for optimal control problems governed by the elliptic homogenization equations,, Comput. Math. Appl., 60 (2010), 390. doi: 10.1016/j.camwa.2010.04.017. Google Scholar

[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. Google Scholar

[41]

A. M. Matache and C. Schwab, Two-scale FEM for homogenization problems,, M2AN Math. Model. Numer. Anal., 36 (2002), 537. doi: 10.1051/m2an:2002025. Google Scholar

[42]

J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems,, Multiscale Model. Simul., 7 (2008), 171. Google Scholar

[43]

J. M. Nordbotten, Adaptive variational multiscale methods for multiphase flow in porous media,, Multiscale Model. Simul., 7 (2008), 1455. Google Scholar

[44]

M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems,, Multiscale Model. Simul., 4 (2005), 88. Google Scholar

[45]

C. Schwab and A.-M. Matache, Generalized {FEM for homogenization problems},, in, 20 (2002), 197. Google Scholar

[46]

B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization,, Netw. Heterog. Media, 6 (2011), 755. Google Scholar

[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. doi: 10.1007/s10409-010-0393-9. Google Scholar

show all references

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. doi: 10.1137/07070108X. Google Scholar

[2]

J. E. Aarnes, Y. Efendiev and L. Jiang, Mixed multiscale finite element methods using limited global information,, Multiscale Model. Simul., 7 (2008), 655. Google Scholar

[3]

A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM,, Multiscale Model. Simul., 4 (2005), 447. Google Scholar

[4]

A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems,, J. Comput. Phys., 191 (2003), 18. doi: 10.1016/S0021-9991(03)00303-6. Google Scholar

[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. doi: 10.1137/0729009. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1090/S0025-5718-2010-02372-5. Google Scholar

[9]

D. Cioranescu and P. Donato, "An Introduction to Homogenization,", The Clarendon Press Oxford University Press, (1999). Google Scholar

[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. doi: 10.1016/j.cma.2008.02.030. Google Scholar

[11]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. Google Scholar

[12]

W. E and B. Engquist, Multiscale modeling and computation,, Notices Amer. Math. Soc., 50 (2003), 1062. Google Scholar

[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. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[14]

Y. Efendiev and T. Hou, Multiscale finite element methods for porous media flows and their applications,, Appl. Numer. Math., 57 (2007), 577. doi: 10.1016/j.apnum.2006.07.009. Google Scholar

[15]

Y. Efendiev and T. Hou, "Multiscale Finite Element Methods,", Surveys and Tutorials in the Applied Mathematical Sciences, 4 (2009). Google Scholar

[16]

Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications,, Commun. Math. Sci., 2 (2004), 553. Google Scholar

[17]

Y. Efendiev and A. Pankov, Numerical homogenization of monotone elliptic operators,, Multiscale Model. Simul., 2 (2003), 62. Google Scholar

[18]

Y. Efendiev and A. Pankov, Numerical homogenization and correctors for nonlinear elliptic equations,, SIAM J. Appl. Math., 65 (2004), 43. doi: 10.1137/S0036139903424886. Google Scholar

[19]

Y. Efendiev and A. Pankov, Numerical homogenization of nonlinear random parabolic operators,, Multiscale Model. Simul., 2 (2004), 237. Google Scholar

[20]

Y. Efendiev and A. Pankov, Homogenization of nonlinear random parabolic operators,, Adv. Differential Equations, 10 (2005), 1235. Google Scholar

[21]

Y. Efendiev and A. Pankov, On homogenization of almost periodic nonlinear parabolic operators,, Int. J. Evol. Equ., 1 (2005), 203. Google Scholar

[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. doi: 10.1016/j.jcp.2010.09.026. Google Scholar

[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. doi: 10.1137/S0036142997330329. Google Scholar

[24]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,", Monographs and Studies in Mathematics, 24 (1985). Google Scholar

[25]

P. Grisvard, "Singularities in Boundary Value Problems,", Recherches en Mathématiques Appliquées, 22 (1992). Google Scholar

[26]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numer. Math., 113 (2009), 601. doi: 10.1007/s00211-009-0244-4. Google Scholar

[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. Google Scholar

[28]

V. H. Hoang, Sparse finite element method for periodic multiscale nonlinear monotone problems,, Multiscale Model. Simul., 7 (2008), 1042. Google Scholar

[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. Google Scholar

[30]

U. Hornung, "Homogenization and Porous Media,", Interdisciplinary Applied Mathematics, 6 (1997). Google Scholar

[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. doi: 10.1006/jcph.1997.5682. Google Scholar

[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. doi: 10.1090/S0025-5718-99-01077-7. Google Scholar

[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. doi: 10.1016/0045-7825(95)00844-9. Google Scholar

[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. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[35]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: duality techniques for elliptic problems,, in, 44 (2005), 181. Google Scholar

[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. doi: 10.1016/j.cma.2006.08.019. Google Scholar

[37]

M. G. Larson and A. Målqvist, An adaptive variational multiscale method for convection-diffusion problems,, Comm. Numer. Methods Engrg., 25 (2009), 65. doi: 10.1002/cnm.1106. Google Scholar

[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. doi: 10.1142/S021820250900370X. Google Scholar

[39]

J. Li, A multiscale finite element method for optimal control problems governed by the elliptic homogenization equations,, Comput. Math. Appl., 60 (2010), 390. doi: 10.1016/j.camwa.2010.04.017. Google Scholar

[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. Google Scholar

[41]

A. M. Matache and C. Schwab, Two-scale FEM for homogenization problems,, M2AN Math. Model. Numer. Anal., 36 (2002), 537. doi: 10.1051/m2an:2002025. Google Scholar

[42]

J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems,, Multiscale Model. Simul., 7 (2008), 171. Google Scholar

[43]

J. M. Nordbotten, Adaptive variational multiscale methods for multiphase flow in porous media,, Multiscale Model. Simul., 7 (2008), 1455. Google Scholar

[44]

M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems,, Multiscale Model. Simul., 4 (2005), 88. Google Scholar

[45]

C. Schwab and A.-M. Matache, Generalized {FEM for homogenization problems},, in, 20 (2002), 197. Google Scholar

[46]

B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization,, Netw. Heterog. Media, 6 (2011), 755. Google Scholar

[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. doi: 10.1007/s10409-010-0393-9. Google Scholar

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